Bearing on a great circle

Percentage Accurate: 79.2% → 99.7%

Time: 26.4s

Alternatives: 29

Speedup: 1.0×

Specification

Math

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

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: 79.2% accurate, 1.0× speedup

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 99.7

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in lambda1 around inf

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

    1. cos-neg-rev N/A

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

    2. associate-*r* N/A

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

    3. cos-neg-rev N/A

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

    4. *-commutative N/A

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

    5. distribute-rgt-out N/A

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

    6. lower-*.f64 N/A

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

    7. lower-cos.f64 N/A

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

    8. cos-neg-rev N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. cos-neg-rev N/A

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

    12. lower-cos.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-sin.f64 N/A

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

    16. lower-sin.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 6: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in lambda1 around 0

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

11. Applied rewrites 99.7%

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

Alternative 7: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2 \cdot \left(-\cos \lambda_1\right), \sin \lambda_2, \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right)}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2 - t\_2}{t\_0 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \lambda_1 - t\_2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2 (* (* (cos lambda1) (sin lambda2)) (cos phi2))))
   (if (<= phi2 -1.9e-8)
     (atan2
      (fma
       (* (cos phi2) (- (cos lambda1)))
       (sin lambda2)
       (* (cos phi2) (* (sin lambda1) (cos lambda2))))
      t_1)
     (if (<= phi2 2.5e-5)
       (atan2
        (- (* (* (cos lambda2) (sin lambda1)) (cos phi2)) t_2)
        (-
         t_0
         (*
          (sin phi1)
          (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
       (atan2 (- (* (* (cos phi2) (cos lambda2)) (sin lambda1)) t_2) t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = (cos(lambda1) * sin(lambda2)) * cos(phi2);
	double tmp;
	if (phi2 <= -1.9e-8) {
		tmp = atan2(fma((cos(phi2) * -cos(lambda1)), sin(lambda2), (cos(phi2) * (sin(lambda1) * cos(lambda2)))), t_1);
	} else if (phi2 <= 2.5e-5) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) * cos(phi2)) - t_2), (t_0 - (sin(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = atan2((((cos(phi2) * cos(lambda2)) * sin(lambda1)) - t_2), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(Float64(cos(lambda1) * sin(lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -1.9e-8)
		tmp = atan(fma(Float64(cos(phi2) * Float64(-cos(lambda1))), sin(lambda2), Float64(cos(phi2) * Float64(sin(lambda1) * cos(lambda2)))), t_1);
	elseif (phi2 <= 2.5e-5)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) * cos(phi2)) - t_2), Float64(t_0 - Float64(sin(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = atan(Float64(Float64(Float64(cos(phi2) * cos(lambda2)) * sin(lambda1)) - t_2), t_1);
	end
	return 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[(t$95$0 - 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[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.9e-8], N[ArcTan[N[(N[(N[Cos[phi2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[phi2, 2.5e-5], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.90000000000000014e-8

   1. Initial program 73.7%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 87.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if -1.90000000000000014e-8 < phi2 < 2.50000000000000012e-5

   1. Initial program 78.7%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. cos-neg-rev N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 2.50000000000000012e-5 < phi2

   1. Initial program 76.7%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.8

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

   6. Applied rewrites 91.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2 \cdot \left(-\cos \lambda_1\right), \sin \lambda_2, \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \lambda_1 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := -\cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2 \cdot t\_2, \sin \lambda_2, \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right)}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(t\_2, \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{t\_0 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1 \cdot \cos \phi_2, \sin \lambda_2, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \lambda_1 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2 (- (cos lambda1))))
   (if (<= phi2 -9.2e-7)
     (atan2
      (fma
       (* (cos phi2) t_2)
       (sin lambda2)
       (* (cos phi2) (* (sin lambda1) (cos lambda2))))
      t_1)
     (if (<= phi2 1.5e-5)
       (atan2
        (fma t_2 (sin lambda2) (* (cos lambda2) (sin lambda1)))
        (-
         t_0
         (*
          (sin phi1)
          (fma
           (* (sin lambda1) (cos phi2))
           (sin lambda2)
           (* (* (cos lambda2) (cos lambda1)) (cos phi2))))))
       (atan2
        (-
         (* (* (cos phi2) (cos lambda2)) (sin lambda1))
         (* (* (cos lambda1) (sin lambda2)) (cos phi2)))
        t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = -cos(lambda1);
	double tmp;
	if (phi2 <= -9.2e-7) {
		tmp = atan2(fma((cos(phi2) * t_2), sin(lambda2), (cos(phi2) * (sin(lambda1) * cos(lambda2)))), t_1);
	} else if (phi2 <= 1.5e-5) {
		tmp = atan2(fma(t_2, sin(lambda2), (cos(lambda2) * sin(lambda1))), (t_0 - (sin(phi1) * fma((sin(lambda1) * cos(phi2)), sin(lambda2), ((cos(lambda2) * cos(lambda1)) * cos(phi2))))));
	} else {
		tmp = atan2((((cos(phi2) * cos(lambda2)) * sin(lambda1)) - ((cos(lambda1) * sin(lambda2)) * cos(phi2))), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(-cos(lambda1))
	tmp = 0.0
	if (phi2 <= -9.2e-7)
		tmp = atan(fma(Float64(cos(phi2) * t_2), sin(lambda2), Float64(cos(phi2) * Float64(sin(lambda1) * cos(lambda2)))), t_1);
	elseif (phi2 <= 1.5e-5)
		tmp = atan(fma(t_2, sin(lambda2), Float64(cos(lambda2) * sin(lambda1))), Float64(t_0 - Float64(sin(phi1) * fma(Float64(sin(lambda1) * cos(phi2)), sin(lambda2), Float64(Float64(cos(lambda2) * cos(lambda1)) * cos(phi2))))));
	else
		tmp = atan(Float64(Float64(Float64(cos(phi2) * cos(lambda2)) * sin(lambda1)) - Float64(Float64(cos(lambda1) * sin(lambda2)) * cos(phi2))), t_1);
	end
	return 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[(t$95$0 - 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[Cos[lambda1], $MachinePrecision])}, If[LessEqual[phi2, -9.2e-7], N[ArcTan[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[phi2, 1.5e-5], N[ArcTan[N[(t$95$2 * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.1999999999999998e-7

   1. Initial program 73.3%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 87.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

   if -9.1999999999999998e-7 < phi2 < 1.50000000000000004e-5

   1. Initial program 78.9%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. cos-neg-rev N/A

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

       8. lower-*.f64 N/A

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

       9. cos-neg-rev N/A

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

       10. lower-cos.f64 N/A

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

       11. lower-sin.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 1.50000000000000004e-5 < phi2

   1. Initial program 76.7%

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

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

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sin-diff N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-sin.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.8

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

   6. Applied rewrites 91.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2 \cdot \left(-\cos \lambda_1\right), \sin \lambda_2, \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\cos \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1 \cdot \cos \phi_2, \sin \lambda_2, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \lambda_1 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.1% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(fma(Float64(cos(phi2) * Float64(-cos(lambda1))), sin(lambda2), Float64(cos(phi2) * Float64(sin(lambda1) * cos(lambda2)))), 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[Cos[phi2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $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 76.7%

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

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

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-sin.f64 89.3

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

4. Applied rewrites 89.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 89.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 89.3

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 89.3

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

6. Applied rewrites 89.3%

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

Alternative 10: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -7.2 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 480000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2, \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (sin phi1) (cos phi2))))
   (if (or (<= lambda1 -7.2e-6) (not (<= lambda1 480000000.0)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1)))
       (cos phi2))
      (- t_0 (* t_1 (cos lambda1))))
     (atan2
      (* (fma (cos lambda2) lambda1 (- (sin lambda2))) (cos phi2))
      (- t_0 (* t_1 (cos (- lambda1 lambda2))))))))

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 tmp;
	if ((lambda1 <= -7.2e-6) || !(lambda1 <= 480000000.0)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	} else {
		tmp = atan2((fma(cos(lambda2), lambda1, -sin(lambda2)) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if ((lambda1 <= -7.2e-6) || !(lambda1 <= 480000000.0))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda1))));
	else
		tmp = atan(Float64(fma(cos(lambda2), lambda1, Float64(-sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	end
	return 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]}, If[Or[LessEqual[lambda1, -7.2e-6], N[Not[LessEqual[lambda1, 480000000.0]], $MachinePrecision]], 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[(t$95$0 - N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Cos[lambda2], $MachinePrecision] * lambda1 + (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -7.19999999999999967e-6 or 4.8e8 < lambda1

   1. Initial program 55.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 55.3

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

   5. Applied rewrites 55.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -7.19999999999999967e-6 < lambda1 < 4.8e8

   1. Initial program 98.2%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.2 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 480000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2, \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.1% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * 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[lambda1], $MachinePrecision] * N[Cos[lambda2], $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 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. 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. *-commutative N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 89.3

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

4. Applied rewrites 89.3%

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

Alternative 12: 89.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (cos lambda2) (sin lambda1)) (* (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((((cos(lambda2) * sin(lambda1)) - (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((((cos(lambda2) * sin(lambda1)) - (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.cos(lambda2) * Math.sin(lambda1)) - (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.cos(lambda2) * math.sin(lambda1)) - (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(cos(lambda2) * sin(lambda1)) - 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((((cos(lambda2) * sin(lambda1)) - (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[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $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 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. 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. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 89.3

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

4. Applied rewrites 89.3%

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

Alternative 13: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\left(-\cos \phi_2\right) \cdot t\_0\right) \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-\sin \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -4.2e-7)
     (atan2
      t_1
      (fma (sin phi2) (cos phi1) (* (* (- (cos phi2)) t_0) (sin phi1))))
     (if (<= phi1 3e-43)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (cos phi2))
        (- (sin phi2) (* (sin phi1) t_0)))
       (atan2
        t_1
        (fma
         (- (sin phi1))
         (* (cos (- lambda2 lambda1)) (cos phi2))
         (* (sin phi2) (cos phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -4.2e-7) {
		tmp = atan2(t_1, fma(sin(phi2), cos(phi1), ((-cos(phi2) * t_0) * sin(phi1))));
	} else if (phi1 <= 3e-43) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (sin(phi1) * t_0)));
	} else {
		tmp = atan2(t_1, fma(-sin(phi1), (cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -4.2e-7)
		tmp = atan(t_1, fma(sin(phi2), cos(phi1), Float64(Float64(Float64(-cos(phi2)) * t_0) * sin(phi1))));
	elseif (phi1 <= 3e-43)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * t_0)));
	else
		tmp = atan(t_1, fma(Float64(-sin(phi1)), Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)), Float64(sin(phi2) * cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.2e-7], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[((-N[Cos[phi2], $MachinePrecision]) * t$95$0), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3e-43], 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[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[((-N[Sin[phi1], $MachinePrecision]) * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.2e-7

   1. Initial program 75.8%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 52.4

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

   5. Applied rewrites 52.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 29.2

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in lambda1 around -inf

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

   10. Applied rewrites 75.9%

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

   if -4.2e-7 < phi1 < 3.00000000000000003e-43

   1. Initial program 73.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 73.6

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

   8. Applied rewrites 73.6%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if 3.00000000000000003e-43 < phi1

   1. Initial program 81.8%

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

   3. Taylor expanded in lambda1 around 0

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

   5. Applied rewrites 81.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\left(-\cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\sin \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{+18} \lor \neg \left(\lambda_1 \leq 3600000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -3.1e+18) (not (<= lambda1 3600000000.0)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -3.1e+18) || !(lambda1 <= 3600000000.0)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	}
	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) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-3.1d+18)) .or. (.not. (lambda1 <= 3600000000.0d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))))
    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 tmp;
	if ((lambda1 <= -3.1e+18) || !(lambda1 <= 3600000000.0)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - ((Math.cos(lambda2) * Math.sin(phi1)) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -3.1e+18) or not (lambda1 <= 3600000000.0):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - ((math.cos(lambda2) * math.sin(phi1)) * math.cos(phi2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -3.1e+18) || !(lambda1 <= 3600000000.0))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -3.1e+18) || ~((lambda1 <= 3600000000.0)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	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]}, If[Or[LessEqual[lambda1, -3.1e+18], N[Not[LessEqual[lambda1, 3600000000.0]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.1e18 or 3.6e9 < lambda1

   1. Initial program 55.4%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 54.8

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

   5. Applied rewrites 54.8%

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

   if -3.1e18 < lambda1 < 3.6e9

   1. Initial program 97.4%

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

   3. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 97.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

   5. Applied rewrites 97.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{+18} \lor \neg \left(\lambda_1 \leq 3600000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.95 \cdot 10^{-195} \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-90}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -1.95e-195) (not (<= lambda2 4.4e-90)))
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- t_0 (* (sin phi1) (cos (- lambda2 lambda1)))))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -1.95e-195) || !(lambda2 <= 4.4e-90)) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	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) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda2 <= (-1.95d-195)) .or. (.not. (lambda2 <= 4.4d-90))) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    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 tmp;
	if ((lambda2 <= -1.95e-195) || !(lambda2 <= 4.4e-90)) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -1.95e-195) or not (lambda2 <= 4.4e-90):
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -1.95e-195) || !(lambda2 <= 4.4e-90))
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda2 <= -1.95e-195) || ~((lambda2 <= 4.4e-90)))
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	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]}, If[Or[LessEqual[lambda2, -1.95e-195], N[Not[LessEqual[lambda2, 4.4e-90]], $MachinePrecision]], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.95e-195 or 4.39999999999999972e-90 < lambda2

   1. Initial program 68.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      5. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right)} \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}} \]

      11. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\lambda_2} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}} \]

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right)} \]

      16. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \color{blue}{\lambda_1}\right)} \]

      17. lower--.f64 57.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}} \]

   5. Applied rewrites 57.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

   if -1.95e-195 < lambda2 < 4.39999999999999972e-90

   1. Initial program 99.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 87.3

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 87.3%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.95 \cdot 10^{-195} \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-90}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.0001:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_2 - t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.00027:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - t\_0 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -0.0001)
     (atan2
      (* (sin (- lambda2)) (cos phi2))
      (- t_2 (* t_0 (cos (- lambda1 lambda2)))))
     (if (<= lambda2 0.00027)
       (atan2 t_1 (- t_2 (* (* (cos lambda1) (cos phi2)) (sin phi1))))
       (atan2 t_1 (- t_2 (* t_0 (cos lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos(phi2);
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_2 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -0.0001) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_2 - (t_0 * cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 0.00027) {
		tmp = atan2(t_1, (t_2 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	} else {
		tmp = atan2(t_1, (t_2 - (t_0 * cos(lambda2))));
	}
	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 = sin((lambda1 - lambda2)) * cos(phi2)
    t_2 = cos(phi1) * sin(phi2)
    if (lambda2 <= (-0.0001d0)) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_2 - (t_0 * cos((lambda1 - lambda2)))))
    else if (lambda2 <= 0.00027d0) then
        tmp = atan2(t_1, (t_2 - ((cos(lambda1) * cos(phi2)) * sin(phi1))))
    else
        tmp = atan2(t_1, (t_2 - (t_0 * cos(lambda2))))
    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.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double t_2 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -0.0001) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_2 - (t_0 * Math.cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 0.00027) {
		tmp = Math.atan2(t_1, (t_2 - ((Math.cos(lambda1) * Math.cos(phi2)) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_1, (t_2 - (t_0 * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.cos(phi2)
	t_1 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	t_2 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -0.0001:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_2 - (t_0 * math.cos((lambda1 - lambda2)))))
	elif lambda2 <= 0.00027:
		tmp = math.atan2(t_1, (t_2 - ((math.cos(lambda1) * math.cos(phi2)) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_1, (t_2 - (t_0 * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -0.0001)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_2 - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda2 <= 0.00027)
		tmp = atan(t_1, Float64(t_2 - Float64(Float64(cos(lambda1) * cos(phi2)) * sin(phi1))));
	else
		tmp = atan(t_1, Float64(t_2 - Float64(t_0 * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * cos(phi2);
	t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	t_2 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -0.0001)
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_2 - (t_0 * cos((lambda1 - lambda2)))));
	elseif (lambda2 <= 0.00027)
		tmp = atan2(t_1, (t_2 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	else
		tmp = atan2(t_1, (t_2 - (t_0 * cos(lambda2))));
	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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -0.0001], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.00027], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$2 - N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.00000000000000005e-4

   1. Initial program 57.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower-neg.f64 60.9

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 60.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if -1.00000000000000005e-4 < lambda2 < 2.70000000000000003e-4

   1. Initial program 99.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 99.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \lambda_1} \]

      3. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

      7. lower-*.f64 99.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

   7. Applied rewrites 99.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if 2.70000000000000003e-4 < lambda2

   1. Initial program 57.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 57.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -72000000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 3600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - t\_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (* (sin phi1) (cos phi2))))
   (if (<= lambda1 -72000000000000.0)
     (atan2 t_1 (- t_0 (* (* (cos lambda1) (cos phi2)) (sin phi1))))
     (if (<= lambda1 3600000000.0)
       (atan2 t_1 (- t_0 (* t_2 (cos lambda2))))
       (atan2
        (* (sin lambda1) (cos phi2))
        (- t_0 (* t_2 (cos (- lambda1 lambda2)))))))))

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 = sin(phi1) * cos(phi2);
	double tmp;
	if (lambda1 <= -72000000000000.0) {
		tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = atan2(t_1, (t_0 - (t_2 * cos(lambda2))));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_2 * cos((lambda1 - lambda2)))));
	}
	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 = sin(phi1) * cos(phi2)
    if (lambda1 <= (-72000000000000.0d0)) then
        tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))))
    else if (lambda1 <= 3600000000.0d0) then
        tmp = atan2(t_1, (t_0 - (t_2 * cos(lambda2))))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_2 * cos((lambda1 - lambda2)))))
    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.sin(phi1) * Math.cos(phi2);
	double tmp;
	if (lambda1 <= -72000000000000.0) {
		tmp = Math.atan2(t_1, (t_0 - ((Math.cos(lambda1) * Math.cos(phi2)) * Math.sin(phi1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = Math.atan2(t_1, (t_0 - (t_2 * Math.cos(lambda2))));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (t_2 * Math.cos((lambda1 - lambda2)))));
	}
	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.sin(phi1) * math.cos(phi2)
	tmp = 0
	if lambda1 <= -72000000000000.0:
		tmp = math.atan2(t_1, (t_0 - ((math.cos(lambda1) * math.cos(phi2)) * math.sin(phi1))))
	elif lambda1 <= 3600000000.0:
		tmp = math.atan2(t_1, (t_0 - (t_2 * math.cos(lambda2))))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (t_2 * math.cos((lambda1 - lambda2)))))
	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 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda1 <= -72000000000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(cos(lambda1) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(t_2 * cos(lambda2))));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(t_2 * cos(Float64(lambda1 - lambda2)))));
	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 = sin(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= -72000000000000.0)
		tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan2(t_1, (t_0 - (t_2 * cos(lambda2))));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_2 * cos((lambda1 - lambda2)))));
	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[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -72000000000000.0], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 3600000000.0], N[ArcTan[t$95$1 / N[(t$95$0 - N[(t$95$2 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -7.2e13

   1. Initial program 55.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 55.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]

   5. Applied rewrites 55.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \lambda_1} \]

      3. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

      7. lower-*.f64 55.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

   7. Applied rewrites 55.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if -7.2e13 < lambda1 < 3.6e9

   1. Initial program 98.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 98.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

   if 3.6e9 < lambda1

   1. Initial program 54.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 54.9

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -72000000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 3600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= lambda1 -72000000000000.0)
     (atan2 t_1 (- t_0 (* (* (cos lambda1) (cos phi2)) (sin phi1))))
     (if (<= lambda1 3600000000.0)
       (atan2 t_1 (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2))))
       (atan2
        (* (sin lambda1) (cos phi2))
        (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))))

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 tmp;
	if (lambda1 <= -72000000000000.0) {
		tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	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) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin((lambda1 - lambda2)) * cos(phi2)
    if (lambda1 <= (-72000000000000.0d0)) then
        tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))))
    else if (lambda1 <= 3600000000.0d0) then
        tmp = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    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 tmp;
	if (lambda1 <= -72000000000000.0) {
		tmp = Math.atan2(t_1, (t_0 - ((Math.cos(lambda1) * Math.cos(phi2)) * Math.sin(phi1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = Math.atan2(t_1, (t_0 - ((Math.cos(lambda2) * Math.sin(phi1)) * Math.cos(phi2))));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	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)
	tmp = 0
	if lambda1 <= -72000000000000.0:
		tmp = math.atan2(t_1, (t_0 - ((math.cos(lambda1) * math.cos(phi2)) * math.sin(phi1))))
	elif lambda1 <= 3600000000.0:
		tmp = math.atan2(t_1, (t_0 - ((math.cos(lambda2) * math.sin(phi1)) * math.cos(phi2))))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	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))
	tmp = 0.0
	if (lambda1 <= -72000000000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(cos(lambda1) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	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);
	tmp = 0.0;
	if (lambda1 <= -72000000000000.0)
		tmp = atan2(t_1, (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	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]}, If[LessEqual[lambda1, -72000000000000.0], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 3600000000.0], 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], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -7.2e13

   1. Initial program 55.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 55.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]

   5. Applied rewrites 55.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \lambda_1} \]

      3. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1}} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

      7. lower-*.f64 55.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

   7. Applied rewrites 55.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if -7.2e13 < lambda1 < 3.6e9

   1. Initial program 98.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)} \cdot \cos \phi_2} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \lambda_2} \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \lambda_2} \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot \cos \phi_2} \]

      7. lower-cos.f64 98.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   if 3.6e9 < lambda1

   1. Initial program 54.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 54.9

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\left(-\cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (fma
   (sin phi2)
   (cos phi1)
   (* (* (- (cos phi2)) (cos (- lambda1 lambda2))) (sin phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi2), cos(phi1), ((-cos(phi2) * cos((lambda1 - lambda2))) * sin(phi1))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi2), cos(phi1), Float64(Float64(Float64(-cos(phi2)) * cos(Float64(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[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
4. Step-by-step derivation

   1. lower-cos.f64 64.6

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]

5. Applied rewrites 64.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   2. flip-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   3. lower-/.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   4. difference-of-squares N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   5. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   6. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   7. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   8. lower-+.f64 35.4

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\lambda_1 + \lambda_2}}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

7. Applied rewrites 35.4%

   \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1 + \lambda_2}\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

8. Taylor expanded in lambda1 around -inf

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-1 \cdot \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right) \cdot \sin \phi_1\right)}} \]

10. Applied rewrites 76.8%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\left(-\cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1\right)}} \]

11. Final simplification 76.8%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\left(-\cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1\right)} \]
12. Add Preprocessing

Alternative 20: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\sin \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (fma
   (- (sin phi1))
   (* (cos (- lambda2 lambda1)) (cos phi2))
   (* (sin phi2) (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(-sin(phi1), (cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(Float64(-sin(phi1)), Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)), Float64(sin(phi2) * cos(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[Sin[phi1], $MachinePrecision]) * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in lambda1 around 0

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Applied rewrites 76.7%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\sin \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}} \]
6. Add Preprocessing

Alternative 21: 66.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda2 lambda1))))))

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((lambda2 - lambda1)))));
}

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((lambda2 - lambda1)))))
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((lambda2 - lambda1)))));
}

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((lambda2 - lambda1)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda2 - lambda1)))));
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[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. cos-neg-rev N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   5. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right)} \]

   7. fp-cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}\right)\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}} \]

   11. remove-double-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\lambda_2} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \]

   12. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)} \]

   13. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}} \]

   14. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right)} \]

   16. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \color{blue}{\lambda_1}\right)} \]

   17. lower--.f64 59.4

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}} \]

5. Applied rewrites 59.4%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]
6. Add Preprocessing

Alternative 22: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (- (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)), (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)), (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.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.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(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)), (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[Sin[phi2], $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 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 58.3

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
6. Add Preprocessing

Alternative 23: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \left(\lambda_2 + \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (sin (* (/ (- lambda1 lambda2) (+ lambda2 lambda1)) (+ lambda2 lambda1)))
   (cos phi2))
  (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((((lambda1 - lambda2) / (lambda2 + lambda1)) * (lambda2 + lambda1))) * cos(phi2)), (sin(phi2) - (sin(phi1) * 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) / (lambda2 + lambda1)) * (lambda2 + lambda1))) * cos(phi2)), (sin(phi2) - (sin(phi1) * 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) / (lambda2 + lambda1)) * (lambda2 + lambda1))) * Math.cos(phi2)), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((((lambda1 - lambda2) / (lambda2 + lambda1)) * (lambda2 + lambda1))) * math.cos(phi2)), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(Float64(Float64(lambda1 - lambda2) / Float64(lambda2 + lambda1)) * Float64(lambda2 + lambda1))) * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((((lambda1 - lambda2) / (lambda2 + lambda1)) * (lambda2 + lambda1))) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] / N[(lambda2 + lambda1), $MachinePrecision]), $MachinePrecision] * N[(lambda2 + lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 58.3

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   7. remove-double-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

   10. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

   13. distribute-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

   14. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

   15. remove-double-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

   16. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

   17. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

   18. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

   20. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   21. lower--.f64 57.5

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

8. Applied rewrites 57.5%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. flip-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. difference-of-squares-rev N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\lambda_1 + \lambda_2}}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. associate-/l* N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\left(\lambda_1 + \lambda_2\right) \cdot \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2} \cdot \left(\lambda_1 + \lambda_2\right)\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2} \cdot \left(\lambda_1 + \lambda_2\right)\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. lower-/.f64 57.6

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\color{blue}{\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}} \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. lift-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\color{blue}{\lambda_1 + \lambda_2}} \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   12. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\color{blue}{\lambda_2 + \lambda_1}} \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   13. lower-+.f64 57.6

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\color{blue}{\lambda_2 + \lambda_1}} \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   14. lift-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \color{blue}{\left(\lambda_1 + \lambda_2\right)}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   15. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \color{blue}{\left(\lambda_2 + \lambda_1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   16. lower-+.f64 57.6

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \color{blue}{\left(\lambda_2 + \lambda_1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

10. Applied rewrites 57.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \left(\lambda_2 + \lambda_1\right)\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
11. Add Preprocessing

Alternative 24: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 2020000\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1 \cdot t\_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -1.9e-5) (not (<= phi2 2020000.0)))
     (atan2 (* t_0 (cos phi2)) (- (sin phi2) (* (sin phi1) (cos lambda1))))
     (atan2
      (* 1.0 t_0)
      (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.9e-5) || !(phi2 <= 2020000.0)) {
		tmp = atan2((t_0 * cos(phi2)), (sin(phi2) - (sin(phi1) * cos(lambda1))));
	} else {
		tmp = atan2((1.0 * t_0), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if ((phi2 <= (-1.9d-5)) .or. (.not. (phi2 <= 2020000.0d0))) then
        tmp = atan2((t_0 * cos(phi2)), (sin(phi2) - (sin(phi1) * cos(lambda1))))
    else
        tmp = atan2((1.0d0 * t_0), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.9e-5) || !(phi2 <= 2020000.0)) {
		tmp = Math.atan2((t_0 * Math.cos(phi2)), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos(lambda1))));
	} else {
		tmp = Math.atan2((1.0 * t_0), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -1.9e-5) or not (phi2 <= 2020000.0):
		tmp = math.atan2((t_0 * math.cos(phi2)), (math.sin(phi2) - (math.sin(phi1) * math.cos(lambda1))))
	else:
		tmp = math.atan2((1.0 * t_0), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -1.9e-5) || !(phi2 <= 2020000.0))
		tmp = atan(Float64(t_0 * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * cos(lambda1))));
	else
		tmp = atan(Float64(1.0 * t_0), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -1.9e-5) || ~((phi2 <= 2020000.0)))
		tmp = atan2((t_0 * cos(phi2)), (sin(phi2) - (sin(phi1) * cos(lambda1))));
	else
		tmp = atan2((1.0 * t_0), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -1.9e-5], N[Not[LessEqual[phi2, 2020000.0]], $MachinePrecision]], N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 * t$95$0), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.9000000000000001e-5 or 2.02e6 < phi2

   1. Initial program 75.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 43.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 43.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 41.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 41.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]

   if -1.9000000000000001e-5 < phi2 < 2.02e6

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 75.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 75.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 75.9

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 75.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       2. distribute-rgt1-in N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       5. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       7. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       8. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       9. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       10. lower-sin.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       11. fp-cancel-sign-sub-inv N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       12. metadata-eval N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       13. *-lft-identity N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       14. lower--.f64 75.7

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Applied rewrites 75.7%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   12. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{1 \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.9%

       \[\leadsto \tan^{-1}_* \frac{1 \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 2020000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.75 \lor \neg \left(\phi_2 \leq 7 \cdot 10^{+31}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
   (if (or (<= phi2 -0.75) (not (<= phi2 7e+31)))
     (atan2 (* (sin lambda1) (cos phi2)) t_0)
     (atan2 (* (fma -0.5 (* phi2 phi2) 1.0) (sin (- lambda1 lambda2))) t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)));
	double tmp;
	if ((phi2 <= -0.75) || !(phi2 <= 7e+31)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), t_0);
	} else {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * sin((lambda1 - lambda2))), t_0);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if ((phi2 <= -0.75) || !(phi2 <= 7e+31))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), t_0);
	else
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * sin(Float64(lambda1 - lambda2))), t_0);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -0.75], N[Not[LessEqual[phi2, 7e+31]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.75 or 7e31 < phi2

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 43.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 42.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 42.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. lower-sin.f64 27.2

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Applied rewrites 27.2%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if -0.75 < phi2 < 7e31

   1. Initial program 77.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 73.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 73.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 73.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 73.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       2. distribute-rgt1-in N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       5. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       7. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       8. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       9. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       10. lower-sin.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       11. fp-cancel-sign-sub-inv N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       12. metadata-eval N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       13. *-lft-identity N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       14. lower--.f64 72.5

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Applied rewrites 72.5%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.75 \lor \neg \left(\phi_2 \leq 7 \cdot 10^{+31}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.000115:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+21}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= lambda2 -0.000115)
     (atan2
      (* (sin (- lambda2)) (cos phi2))
      (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))
     (if (<= lambda2 1.4e+21)
       (atan2 t_0 (- (sin phi2) (* (sin phi1) (cos lambda1))))
       (atan2 t_0 (- (sin phi2) (* (sin phi1) (cos lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (lambda2 <= -0.000115) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 1.4e+21) {
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))));
	} else {
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))));
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2)) * cos(phi2)
    if (lambda2 <= (-0.000115d0)) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else if (lambda2 <= 1.4d+21) then
        tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))))
    else
        tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double tmp;
	if (lambda2 <= -0.000115) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 1.4e+21) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.sin(phi1) * Math.cos(lambda1))));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.sin(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	tmp = 0
	if lambda2 <= -0.000115:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	elif lambda2 <= 1.4e+21:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.sin(phi1) * math.cos(lambda1))))
	else:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.sin(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -0.000115)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda2 <= 1.4e+21)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(sin(phi1) * cos(lambda1))));
	else
		tmp = atan(t_0, Float64(sin(phi2) - Float64(sin(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	tmp = 0.0;
	if (lambda2 <= -0.000115)
		tmp = atan2((sin(-lambda2) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	elseif (lambda2 <= 1.4e+21)
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))));
	else
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -0.000115], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 1.4e+21], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.15e-4

   1. Initial program 57.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 46.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 46.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       2. lower-neg.f64 48.9

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Applied rewrites 48.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if -1.15e-4 < lambda2 < 1.4e21

   1. Initial program 97.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 72.3

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 72.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 71.0

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 71.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]

   if 1.4e21 < lambda2

   1. Initial program 58.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 44.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 44.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 44.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 44.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_1 \leq 3600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= lambda1 -5e+20)
     (atan2 t_0 (- (sin phi2) (* (sin phi1) (cos lambda1))))
     (if (<= lambda1 3600000000.0)
       (atan2 t_0 (- (sin phi2) (* (sin phi1) (cos lambda2))))
       (atan2
        (* (sin lambda1) (cos phi2))
        (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (lambda1 <= -5e+20) {
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2)) * cos(phi2)
    if (lambda1 <= (-5d+20)) then
        tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))))
    else if (lambda1 <= 3600000000.0d0) then
        tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double tmp;
	if (lambda1 <= -5e+20) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.sin(phi1) * Math.cos(lambda1))));
	} else if (lambda1 <= 3600000000.0) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.sin(phi1) * Math.cos(lambda2))));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	tmp = 0
	if lambda1 <= -5e+20:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.sin(phi1) * math.cos(lambda1))))
	elif lambda1 <= 3600000000.0:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.sin(phi1) * math.cos(lambda2))))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (lambda1 <= -5e+20)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(sin(phi1) * cos(lambda1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(sin(phi1) * cos(lambda2))));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= -5e+20)
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda1))));
	elseif (lambda1 <= 3600000000.0)
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos(lambda2))));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -5e+20], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 3600000000.0], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -5e20

   1. Initial program 57.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 45.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 45.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 45.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 45.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1} \]

   if -5e20 < lambda1 < 3.6e9

   1. Initial program 96.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 72.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 72.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 70.8

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 70.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_2} \]

   if 3.6e9 < lambda1

   1. Initial program 54.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 42.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 42.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

      9. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

      10. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

      14. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

      16. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

      17. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

      20. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      21. lower--.f64 42.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   8. Applied rewrites 42.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. lower-sin.f64 42.8

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Applied rewrites 42.8%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - (sin(phi1) * 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)), (sin(phi2) - (sin(phi1) * 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.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - (sin(phi1) * 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[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 58.3

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   7. remove-double-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

   10. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

   13. distribute-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

   14. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

   15. remove-double-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

   16. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

   17. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

   18. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

   20. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   21. lower--.f64 57.5

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

8. Applied rewrites 57.5%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]
9. Add Preprocessing

Alternative 29: 48.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* 1.0 (sin (- lambda1 lambda2)))
  (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((1.0 * sin((lambda1 - lambda2))), (sin(phi2) - (sin(phi1) * 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((1.0d0 * sin((lambda1 - lambda2))), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((1.0 * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((1.0 * math.sin((lambda1 - lambda2))), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(1.0 * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((1.0 * sin((lambda1 - lambda2))), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(1.0 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 58.3

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   7. remove-double-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}} \]

   10. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}} \]

   13. distribute-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right)\right)}} \]

   14. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)} \]

   15. remove-double-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1}\right)} \]

   16. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}} \]

   17. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)} \]

   18. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)} \]

   20. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   21. lower--.f64 57.5

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

8. Applied rewrites 57.5%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    4. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    5. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    7. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    8. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    9. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    10. lower-sin.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    11. fp-cancel-sign-sub-inv N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    13. *-lft-identity N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

    14. lower--.f64 40.8

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

11. Applied rewrites 40.8%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

12. Taylor expanded in phi2 around 0

    \[\leadsto \tan^{-1}_* \frac{1 \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
13. Step-by-step derivation

14. Applied rewrites 42.3%

    \[\leadsto \tan^{-1}_* \frac{1 \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024353 
(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))))))