Bearing on a great circle

Percentage Accurate: 79.2% → 99.7%

Time: 21.4s

Alternatives: 40

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. sin-sum N/A

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

   6. cos-neg-rev N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 89.2%

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

3. Applied rewrites 89.2%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.7%

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. sin-sum N/A

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

   6. cos-neg-rev N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 89.2%

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

3. Applied rewrites 89.2%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.7%

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

5. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-*.f64 99.7%

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

7. Applied rewrites 99.7%

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

Alternative 3: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - t\_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (*
         (fma
          (- (sin lambda2))
          (cos lambda1)
          (* (sin lambda1) (cos lambda2)))
         (cos phi2)))
       (t_1 (* (sin phi1) (cos phi2)))
       (t_2
        (atan2
         t_0
         (-
          (* (cos phi1) (sin phi2))
          (* t_1 (cos (- lambda1 lambda2)))))))
  (if (<= phi2 -1e-14)
    t_2
    (if (<= phi2 5.3e-52)
      (atan2
       t_0
       (-
        (sin phi2)
        (*
         t_1
         (fma
          (sin lambda2)
          (sin lambda1)
          (* (cos lambda2) (cos lambda1))))))
      t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double t_2 = atan2(t_0, ((cos(phi1) * sin(phi2)) - (t_1 * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -1e-14) {
		tmp = t_2;
	} else if (phi2 <= 5.3e-52) {
		tmp = atan2(t_0, (sin(phi2) - (t_1 * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_1 * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi2 <= -1e-14)
		tmp = t_2;
	elseif (phi2 <= 5.3e-52)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(t_1 * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1e-14], t$95$2, If[LessEqual[phi2, 5.3e-52], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$1 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1e-14 or 5.3000000000000003e-52 < phi2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   if -1e-14 < phi2 < 5.3000000000000003e-52

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 81.0%

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

   8. Applied rewrites 81.0%

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

Alternative 4: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_1}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -9000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1
        (*
         (fma
          (- (sin lambda2))
          (cos lambda1)
          (* (sin lambda1) (cos lambda2)))
         (cos phi2)))
       (t_2
        (atan2
         t_1
         (-
          t_0
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
  (if (<= phi2 -9000000.0)
    t_2
    (if (<= phi2 5.3e-52)
      (atan2
       t_1
       (-
        t_0
        (*
         (sin phi1)
         (fma
          (cos lambda1)
          (cos lambda2)
          (* (sin lambda1) (sin lambda2))))))
      t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2);
	double t_2 = atan2(t_1, (t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -9000000.0) {
		tmp = t_2;
	} else if (phi2 <= 5.3e-52) {
		tmp = atan2(t_1, (t_0 - (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2))
	t_2 = atan(t_1, Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi2 <= -9000000.0)
		tmp = t_2;
	elseif (phi2 <= 5.3e-52)
		tmp = atan(t_1, Float64(t_0 - Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = t_2;
	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[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[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]], $MachinePrecision]}, If[LessEqual[phi2, -9000000.0], t$95$2, If[LessEqual[phi2, 5.3e-52], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9e6 or 5.3000000000000003e-52 < phi2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   if -9e6 < phi2 < 5.3000000000000003e-52

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 81.3%

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

   8. Applied rewrites 81.3%

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

Alternative 5: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{-1 \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (*
         (fma
          (- (sin lambda2))
          (cos lambda1)
          (* (sin lambda1) (cos lambda2)))
         (cos phi2)))
       (t_1
        (atan2
         t_0
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
  (if (<= phi2 -1.8e-12)
    t_1
    (if (<= phi2 5.3e-52)
      (atan2
       t_0
       (*
        -1.0
        (*
         (sin phi1)
         (fma
          (cos lambda1)
          (cos lambda2)
          (* (sin lambda1) (sin lambda2))))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2);
	double t_1 = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -1.8e-12) {
		tmp = t_1;
	} else if (phi2 <= 5.3e-52) {
		tmp = atan2(t_0, (-1.0 * (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2))
	t_1 = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi2 <= -1.8e-12)
		tmp = t_1;
	elseif (phi2 <= 5.3e-52)
		tmp = atan(t_0, Float64(-1.0 * Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1.8e-12], t$95$1, If[LessEqual[phi2, 5.3e-52], N[ArcTan[t$95$0 / N[(-1.0 * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.8e-12 or 5.3000000000000003e-52 < phi2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   if -1.8e-12 < phi2 < 5.3000000000000003e-52

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 58.5%

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

   8. Applied rewrites 58.5%

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

Alternative 6: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \tan^{-1}_* \frac{\left(t\_0 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t\_0\right) \cdot \cos \phi_2}{-1 \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (sin lambda1) (cos lambda2)))
       (t_1
        (atan2
         (* (- t_0 (* (sin lambda2) (cos lambda1))) (cos phi2))
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
  (if (<= phi2 -1.8e-12)
    t_1
    (if (<= phi2 5.3e-52)
      (atan2
       (* (fma (- (sin lambda2)) (cos lambda1) t_0) (cos phi2))
       (*
        -1.0
        (*
         (sin phi1)
         (fma
          (cos lambda1)
          (cos lambda2)
          (* (sin lambda1) (sin lambda2))))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = atan2(((t_0 - (sin(lambda2) * cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -1.8e-12) {
		tmp = t_1;
	} else if (phi2 <= 5.3e-52) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), t_0) * cos(phi2)), (-1.0 * (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = atan(Float64(Float64(t_0 - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi2 <= -1.8e-12)
		tmp = t_1;
	elseif (phi2 <= 5.3e-52)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), t_0) * cos(phi2)), Float64(-1.0 * Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[(t$95$0 - 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]}, If[LessEqual[phi2, -1.8e-12], t$95$1, If[LessEqual[phi2, 5.3e-52], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.8e-12 or 5.3000000000000003e-52 < phi2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   if -1.8e-12 < phi2 < 5.3000000000000003e-52

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 58.5%

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

   8. Applied rewrites 58.5%

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

Alternative 7: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t\_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -5.1 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{t\_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1 (- t_0 (* (cos lambda2) (* (cos phi2) (sin phi1))))))
  (if (<= lambda2 -5.1e+17)
    (atan2
     (*
      (fma
       (- (sin lambda2))
       (cos lambda1)
       (* (sin lambda1) (cos lambda2)))
      (cos phi2))
     t_1)
    (if (<= lambda2 1.35e+46)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (-
        t_0
        (*
         (* (sin phi1) (cos phi2))
         (fma
          (sin lambda2)
          (sin lambda1)
          (* (cos lambda2) (cos lambda1))))))
      (atan2
       (*
        (-
         (* (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 - (cos(lambda2) * (cos(phi2) * sin(phi1)));
	double tmp;
	if (lambda2 <= -5.1e+17) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), t_1);
	} else if (lambda2 <= 1.35e+46) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2((((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(cos(lambda2) * Float64(cos(phi2) * sin(phi1))))
	tmp = 0.0
	if (lambda2 <= -5.1e+17)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), t_1);
	elseif (lambda2 <= 1.35e+46)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - 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[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -5.1e+17], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[lambda2, 1.35e+46], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -5.1e17

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

   if -5.1e17 < lambda2 < 1.3500000000000001e46

   1. Initial program 79.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 79.4%

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

   3. Applied rewrites 79.4%

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

   if 1.3500000000000001e46 < lambda2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 79.2%

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

   10. Applied rewrites 79.2%

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

Alternative 8: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -5.1 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{t\_0}\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (-
         (* (cos phi1) (sin phi2))
         (* (cos lambda2) (* (cos phi2) (sin phi1))))))
  (if (<= lambda2 -5.1e+17)
    (atan2
     (*
      (fma
       (- (sin lambda2))
       (cos lambda1)
       (* (sin lambda1) (cos lambda2)))
      (cos phi2))
     t_0)
    (if (<= lambda2 5.8e+46)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (fma
        (sin phi1)
        (- (* (cos (- lambda2 lambda1)) (cos phi2)))
        (* (sin phi2) (cos phi1))))
      (atan2
       (*
        (-
         (* (cos lambda2) (sin lambda1))
         (* (cos lambda1) (sin lambda2)))
        (cos phi2))
       t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(phi1) * sin(phi2)) - (cos(lambda2) * (cos(phi2) * sin(phi1)));
	double tmp;
	if (lambda2 <= -5.1e+17) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), t_0);
	} else if (lambda2 <= 5.8e+46) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), -(cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	} else {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), t_0);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1))))
	tmp = 0.0
	if (lambda2 <= -5.1e+17)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), t_0);
	elseif (lambda2 <= 5.8e+46)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))), Float64(sin(phi2) * cos(phi1))));
	else
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), t_0);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -5.1e+17], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], If[LessEqual[lambda2, 5.8e+46], 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], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -5.1e17

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

   if -5.1e17 < lambda2 < 5.8000000000000004e46

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 79.2%

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

   if 5.8000000000000004e46 < lambda2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 79.2%

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

   10. Applied rewrites 79.2%

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

Alternative 9: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -5.1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (atan2
         (*
          (-
           (* (cos lambda2) (sin lambda1))
           (* (cos lambda1) (sin lambda2)))
          (cos phi2))
         (-
          (* (cos phi1) (sin phi2))
          (* (cos lambda2) (* (cos phi2) (sin phi1)))))))
  (if (<= lambda2 -5.1e+17)
    t_0
    (if (<= lambda2 5.8e+46)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (fma
        (sin phi1)
        (- (* (cos (- lambda2 lambda1)) (cos phi2)))
        (* (sin phi2) (cos phi1))))
      t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	double tmp;
	if (lambda2 <= -5.1e+17) {
		tmp = t_0;
	} else if (lambda2 <= 5.8e+46) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), -(cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))))
	tmp = 0.0
	if (lambda2 <= -5.1e+17)
		tmp = t_0;
	elseif (lambda2 <= 5.8e+46)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))), Float64(sin(phi2) * cos(phi1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -5.1e+17], t$95$0, If[LessEqual[lambda2, 5.8e+46], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.1e17 or 5.8000000000000004e46 < lambda2

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 79.2%

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

   10. Applied rewrites 79.2%

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

   if -5.1e17 < lambda2 < 5.8000000000000004e46

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 79.2%

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

Alternative 10: 88.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 89.2%

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

3. Applied rewrites 89.2%

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

Alternative 11: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := -\sin \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_3 := \tan^{-1}_* \frac{\mathsf{fma}\left(t\_0, 1, t\_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_1}\\ \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 0.0215:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(t\_0, \cos \lambda_1, t\_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (sin lambda2)))
       (t_1 (cos (- lambda1 lambda2)))
       (t_2 (* (sin lambda1) (cos lambda2)))
       (t_3
        (atan2
         (* (fma t_0 1.0 t_2) (cos phi2))
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) t_1)))))
  (if (<= phi1 -1.25e-5)
    t_3
    (if (<= phi1 0.0215)
      (atan2
       (* (fma t_0 (cos lambda1) t_2) (cos phi2))
       (+ (sin phi2) (* -1.0 (* phi1 (* (cos phi2) t_1)))))
      t_3))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -sin(lambda2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin(lambda1) * cos(lambda2);
	double t_3 = atan2((fma(t_0, 1.0, t_2) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_1)));
	double tmp;
	if (phi1 <= -1.25e-5) {
		tmp = t_3;
	} else if (phi1 <= 0.0215) {
		tmp = atan2((fma(t_0, cos(lambda1), t_2) * cos(phi2)), (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * t_1)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-sin(lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(sin(lambda1) * cos(lambda2))
	t_3 = atan(Float64(fma(t_0, 1.0, t_2) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * t_1)))
	tmp = 0.0
	if (phi1 <= -1.25e-5)
		tmp = t_3;
	elseif (phi1 <= 0.0215)
		tmp = atan(Float64(fma(t_0, cos(lambda1), t_2) * cos(phi2)), Float64(sin(phi2) + Float64(-1.0 * Float64(phi1 * Float64(cos(phi2) * t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = (-N[Sin[lambda2], $MachinePrecision])}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(t$95$0 * 1.0 + t$95$2), $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] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.25e-5], t$95$3, If[LessEqual[phi1, 0.0215], N[ArcTan[N[(N[(t$95$0 * N[Cos[lambda1], $MachinePrecision] + t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] + N[(-1.0 * N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.2500000000000001e-5 or 0.021499999999999998 < phi1

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in lambda1 around 0

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

   6. Applied rewrites 80.7%

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

   if -1.2500000000000001e-5 < phi1 < 0.021499999999999998

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 57.1%

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

   6. Applied rewrites 57.1%

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

Alternative 12: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 52:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (atan2
         (* (sin (- lambda1 lambda2)) (cos phi2))
         (fma
          (sin phi1)
          (- (* (cos (- lambda2 lambda1)) (cos phi2)))
          (* (sin phi2) (cos phi1))))))
  (if (<= phi1 -4.8e-5)
    t_0
    (if (<= phi1 52.0)
      (atan2
       (*
        (fma
         (- (sin lambda2))
         (cos lambda1)
         (* (sin lambda1) (cos lambda2)))
        (cos phi2))
       (- (sin phi2) (* (cos lambda2) (* (cos phi2) (sin phi1)))))
      t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), -(cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	double tmp;
	if (phi1 <= -4.8e-5) {
		tmp = t_0;
	} else if (phi1 <= 52.0) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), (sin(phi2) - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))), Float64(sin(phi2) * cos(phi1))))
	tmp = 0.0
	if (phi1 <= -4.8e-5)
		tmp = t_0;
	elseif (phi1 <= 52.0)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), Float64(sin(phi2) - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, If[LessEqual[phi1, -4.8e-5], t$95$0, If[LessEqual[phi1, 52.0], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.8000000000000001e-5 or 52 < phi1

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 79.2%

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

   if -4.8000000000000001e-5 < phi1 < 52

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 79.2%

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in phi1 around 0

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

       1. lower-sin.f64 70.3%

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

   11. Applied rewrites 70.3%

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

Alternative 13: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 14.5:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (atan2
         (* (sin (- lambda1 lambda2)) (cos phi2))
         (fma
          (sin phi1)
          (- (* (cos (- lambda2 lambda1)) (cos phi2)))
          (* (sin phi2) (cos phi1))))))
  (if (<= phi1 -4.8e-5)
    t_0
    (if (<= phi1 14.5)
      (atan2
       (*
        (fma
         (- (sin lambda2))
         (cos lambda1)
         (* (sin lambda1) (cos lambda2)))
        (cos phi2))
       (+
        (sin phi2)
        (* -1.0 (* phi1 (* (cos phi2) (cos (- lambda1 lambda2)))))))
      t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), -(cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	double tmp;
	if (phi1 <= -4.8e-5) {
		tmp = t_0;
	} else if (phi1 <= 14.5) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * cos((lambda1 - lambda2)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))), Float64(sin(phi2) * cos(phi1))))
	tmp = 0.0
	if (phi1 <= -4.8e-5)
		tmp = t_0;
	elseif (phi1 <= 14.5)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), Float64(sin(phi2) + Float64(-1.0 * Float64(phi1 * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, If[LessEqual[phi1, -4.8e-5], t$95$0, If[LessEqual[phi1, 14.5], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] + N[(-1.0 * N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.8000000000000001e-5 or 14.5 < phi1

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 79.2%

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

   if -4.8000000000000001e-5 < phi1 < 14.5

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 57.1%

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

   6. Applied rewrites 57.1%

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

Alternative 14: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -10000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_1, -t\_0, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 400000000:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \sin \phi_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos (- lambda2 lambda1)) (cos phi2)))
       (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
  (if (<= phi2 -10000.0)
    (atan2 t_1 (fma (sin phi1) (- t_0) (* (sin phi2) (cos phi1))))
    (if (<= phi2 400000000.0)
      (atan2
       (*
        (fma
         (- (sin lambda2))
         (cos lambda1)
         (* (sin lambda1) (cos lambda2)))
        (cos phi2))
       (-
        (* phi2 (cos phi1))
        (* (cos (- lambda1 lambda2)) (sin phi1))))
      (atan2 t_1 (- (* (cos phi1) (sin phi2)) (* t_0 (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1)) * cos(phi2);
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi2 <= -10000.0) {
		tmp = atan2(t_1, fma(sin(phi1), -t_0, (sin(phi2) * cos(phi1))));
	} else if (phi2 <= 400000000.0) {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), ((phi2 * cos(phi1)) - (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (t_0 * sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -10000.0)
		tmp = atan(t_1, fma(sin(phi1), Float64(-t_0), Float64(sin(phi2) * cos(phi1))));
	elseif (phi2 <= 400000000.0)
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), Float64(Float64(phi2 * cos(phi1)) - Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * sin(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -10000.0], N[ArcTan[t$95$1 / N[(N[Sin[phi1], $MachinePrecision] * (-t$95$0) + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 400000000.0], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1e4

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 79.2%

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

   if -1e4 < phi2 < 4e8

   1. Initial program 79.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.2%

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      6. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      7. lower-sin.f64 52.7%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   6. Applied rewrites 52.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   if 4e8 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1}} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

      7. lower-*.f64 79.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      9. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      10. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      11. lift--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      12. sub-negate-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      13. lower--.f64 79.2%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   3. Applied rewrites 79.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-1, \sin \lambda_2, \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (if (<= lambda2 1.35e+46)
  (atan2
   (* (sin (- lambda1 lambda2)) (cos phi2))
   (fma
    (sin phi1)
    (- (* (cos (- lambda2 lambda1)) (cos phi2)))
    (* (sin phi2) (cos phi1))))
  (atan2
   (* (fma -1.0 (sin lambda2) (* lambda1 (cos lambda2))) (cos phi2))
   (-
    (* (cos phi1) (sin phi2))
    (* (cos lambda2) (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.35e+46) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), -(cos((lambda2 - lambda1)) * cos(phi2)), (sin(phi2) * cos(phi1))));
	} else {
		tmp = atan2((fma(-1.0, sin(lambda2), (lambda1 * cos(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.35e+46)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(sin(phi1), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2))), Float64(sin(phi2) * cos(phi1))));
	else
		tmp = atan(Float64(fma(-1.0, sin(lambda2), Float64(lambda1 * cos(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.35e+46], 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], N[ArcTan[N[(N[(-1.0 * N[Sin[lambda2], $MachinePrecision] + N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.3500000000000001e46

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      7. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   3. Applied rewrites 79.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}} \]

   if 1.3500000000000001e46 < lambda2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sub-flip N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. sin-sum N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\cos \lambda_2} \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\sin \lambda_2\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{-\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\color{blue}{\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-cos.f64 89.2%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      3. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

      12. lower-*.f64 99.7%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}} \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 79.2%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   8. Applied rewrites 79.2%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(-1 \cdot \sin \lambda_2 + \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \color{blue}{\sin \lambda_2}, \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

       2. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \sin \lambda_2, \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \sin \lambda_2, \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

       4. lower-cos.f64 59.6%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \sin \lambda_2, \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   11. Applied rewrites 59.6%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-1, \sin \lambda_2, \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \lambda_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.24:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 0.21:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1 (* (sin lambda1) (cos phi2))))
  (if (<= lambda1 -0.24)
    (atan2
     t_1
     (-
      (* (sin phi2) (cos phi1))
      (* (* (cos (- lambda2 lambda1)) (cos phi2)) (sin phi1))))
    (if (<= lambda1 0.21)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (- t_0 (* (cos phi2) (* (cos (- lambda2)) (sin phi1)))))
      (atan2
       t_1
       (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(lambda1) * cos(phi2);
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = atan2(t_1, ((sin(phi2) * cos(phi1)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	} else if (lambda1 <= 0.21) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos(phi2) * (cos(-lambda2) * sin(phi1)))));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	}
	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) * cos(phi2)
    if (lambda1 <= (-0.24d0)) then
        tmp = atan2(t_1, ((sin(phi2) * cos(phi1)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))))
    else if (lambda1 <= 0.21d0) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos(phi2) * (cos(-lambda2) * sin(phi1)))))
    else
        tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    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) * Math.cos(phi2);
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = Math.atan2(t_1, ((Math.sin(phi2) * Math.cos(phi1)) - ((Math.cos((lambda2 - lambda1)) * Math.cos(phi2)) * Math.sin(phi1))));
	} else if (lambda1 <= 0.21) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (Math.cos(phi2) * (Math.cos(-lambda2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin(lambda1) * math.cos(phi2)
	tmp = 0
	if lambda1 <= -0.24:
		tmp = math.atan2(t_1, ((math.sin(phi2) * math.cos(phi1)) - ((math.cos((lambda2 - lambda1)) * math.cos(phi2)) * math.sin(phi1))))
	elif lambda1 <= 0.21:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (math.cos(phi2) * (math.cos(-lambda2) * math.sin(phi1)))))
	else:
		tmp = math.atan2(t_1, (t_0 - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(lambda1) * cos(phi2))
	tmp = 0.0
	if (lambda1 <= -0.24)
		tmp = atan(t_1, Float64(Float64(sin(phi2) * cos(phi1)) - Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 0.21)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(cos(Float64(-lambda2)) * sin(phi1)))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin(lambda1) * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.24)
		tmp = atan2(t_1, ((sin(phi2) * cos(phi1)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 0.21)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos(phi2) * (cos(-lambda2) * sin(phi1)))));
	else
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	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[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.24], N[ArcTan[t$95$1 / N[(N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 0.21], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[(-lambda2)], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.23999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\cos \phi_1} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_1\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\frac{\color{blue}{\pi}}{2} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\color{blue}{\pi \cdot \frac{1}{2}} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\pi \cdot \color{blue}{\frac{1}{2}} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\color{blue}{\frac{1}{2} \cdot \pi} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-fma.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \phi_1\right)\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \phi_1\right)\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if -0.23999999999999999 < lambda1 < 0.20999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)} \]

      6. lower-sin.f64 69.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 69.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]

   if 0.20999999999999999 < lambda1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\lambda_1 \leq -0.24:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{t\_1 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 0.21:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (sin lambda1) (cos phi2)))
       (t_1 (* (sin phi2) (cos phi1))))
  (if (<= lambda1 -0.24)
    (atan2
     t_0
     (- t_1 (* (* (cos (- lambda2 lambda1)) (cos phi2)) (sin phi1))))
    (if (<= lambda1 0.21)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (- t_1 (* (* (sin phi1) (cos phi2)) (cos lambda2))))
      (atan2
       t_0
       (-
        (* (cos phi1) (sin phi2))
        (* (cos lambda1) (* (cos phi2) (sin phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(phi2);
	double t_1 = sin(phi2) * cos(phi1);
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = atan2(t_0, (t_1 - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	} else if (lambda1 <= 0.21) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	}
	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 = sin(lambda1) * cos(phi2)
    t_1 = sin(phi2) * cos(phi1)
    if (lambda1 <= (-0.24d0)) then
        tmp = atan2(t_0, (t_1 - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))))
    else if (lambda1 <= 0.21d0) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda2))))
    else
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    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) * Math.cos(phi2);
	double t_1 = Math.sin(phi2) * Math.cos(phi1);
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = Math.atan2(t_0, (t_1 - ((Math.cos((lambda2 - lambda1)) * Math.cos(phi2)) * Math.sin(phi1))));
	} else if (lambda1 <= 0.21) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_1 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(lambda1) * math.cos(phi2)
	t_1 = math.sin(phi2) * math.cos(phi1)
	tmp = 0
	if lambda1 <= -0.24:
		tmp = math.atan2(t_0, (t_1 - ((math.cos((lambda2 - lambda1)) * math.cos(phi2)) * math.sin(phi1))))
	elif lambda1 <= 0.21:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_1 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	else:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(phi2))
	t_1 = Float64(sin(phi2) * cos(phi1))
	tmp = 0.0
	if (lambda1 <= -0.24)
		tmp = atan(t_0, Float64(t_1 - Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 0.21)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_1 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(lambda1) * cos(phi2);
	t_1 = sin(phi2) * cos(phi1);
	tmp = 0.0;
	if (lambda1 <= -0.24)
		tmp = atan2(t_0, (t_1 - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
	elseif (lambda1 <= 0.21)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.24], N[ArcTan[t$95$0 / N[(t$95$1 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 0.21], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.23999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\cos \phi_1} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_1\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\frac{\color{blue}{\pi}}{2} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\color{blue}{\pi \cdot \frac{1}{2}} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\pi \cdot \color{blue}{\frac{1}{2}} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \left(\color{blue}{\frac{1}{2} \cdot \pi} + \phi_1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-fma.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \phi_1\right)\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \phi_1\right)\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if -0.23999999999999999 < lambda1 < 0.20999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sub-flip N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. sin-sum N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\cos \lambda_2} \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\sin \lambda_2\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{-\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\color{blue}{\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-cos.f64 89.2%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      3. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

      12. lower-*.f64 99.7%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}} \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 79.2%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   8. Applied rewrites 79.2%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}} \]

   if 0.20999999999999999 < lambda1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.24:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 0.21:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (atan2
         (* (sin lambda1) (cos phi2))
         (-
          (* (cos phi1) (sin phi2))
          (* (cos lambda1) (* (cos phi2) (sin phi1)))))))
  (if (<= lambda1 -0.24)
    t_0
    (if (<= lambda1 0.21)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (-
        (* (sin phi2) (cos phi1))
        (* (* (sin phi1) (cos phi2)) (cos lambda2))))
      t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = t_0;
	} else if (lambda1 <= 0.21) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((sin(phi2) * cos(phi1)) - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    if (lambda1 <= (-0.24d0)) then
        tmp = t_0
    else if (lambda1 <= 0.21d0) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((sin(phi2) * cos(phi1)) - ((sin(phi1) * cos(phi2)) * cos(lambda2))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	double tmp;
	if (lambda1 <= -0.24) {
		tmp = t_0;
	} else if (lambda1 <= 0.21) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.sin(phi2) * Math.cos(phi1)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2((math.sin(lambda1) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	tmp = 0
	if lambda1 <= -0.24:
		tmp = t_0
	elif lambda1 <= 0.21:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.sin(phi2) * math.cos(phi1)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(sin(lambda1) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))))
	tmp = 0.0
	if (lambda1 <= -0.24)
		tmp = t_0;
	elseif (lambda1 <= 0.21)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(sin(phi2) * cos(phi1)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	tmp = 0.0;
	if (lambda1 <= -0.24)
		tmp = t_0;
	elseif (lambda1 <= 0.21)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((sin(phi2) * cos(phi1)) - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -0.24], t$95$0, If[LessEqual[lambda1, 0.21], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.23999999999999999 or 0.20999999999999999 < lambda1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

   if -0.23999999999999999 < lambda1 < 0.20999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sub-flip N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. sin-sum N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\cos \lambda_2} \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\sin \lambda_2\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{-\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\color{blue}{\sin \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-cos.f64 89.2%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      3. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

      12. lower-*.f64 99.7%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}} \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 79.2%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   8. Applied rewrites 79.2%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.0% accurate, 1.0× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)} \]

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(sin(phi1), Float64(-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 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   4. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

   6. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

   7. associate-*l* N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

3. Applied rewrites 79.2%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, -\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}} \]
4. Add Preprocessing

Alternative 20: 79.0% accurate, 1.0× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (-
  (* (cos phi1) (sin phi2))
  (* (* (cos (- lambda2 lambda1)) (cos phi2)) (sin phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos((lambda2 - lambda1)) * Math.cos(phi2)) * Math.sin(phi1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos((lambda2 - lambda1)) * math.cos(phi2)) * math.sin(phi1))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)) * sin(phi1))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. associate-*l* N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1}} \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \phi_1}} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

   7. lower-*.f64 79.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot \sin \phi_1} \]

   8. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   9. cos-neg-rev N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   10. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   11. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   12. sub-negate-rev N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   13. lower--.f64 79.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

3. Applied rewrites 79.2%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]
4. Add Preprocessing

Alternative 21: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(-\lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - t\_2}\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t\_1 + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + -0.5 \cdot \left(\lambda_1 \cdot t\_1\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot t\_2}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1 (sin (- lambda2)))
       (t_2 (* (cos (- lambda1 lambda2)) (sin phi1)))
       (t_3 (* (sin (- lambda1 lambda2)) (cos phi2))))
  (if (<= lambda2 -5.5e+66)
    (atan2 t_3 (- t_0 t_2))
    (if (<= lambda2 5.8e+46)
      (atan2 t_3 (- t_0 (* (* (sin phi1) (cos phi2)) (cos lambda1))))
      (atan2
       (*
        (+
         t_1
         (* lambda1 (+ (cos (- lambda2)) (* -0.5 (* lambda1 t_1)))))
        (cos phi2))
       (* -1.0 t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(-lambda2);
	double t_2 = cos((lambda1 - lambda2)) * sin(phi1);
	double t_3 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (lambda2 <= -5.5e+66) {
		tmp = atan2(t_3, (t_0 - t_2));
	} else if (lambda2 <= 5.8e+46) {
		tmp = atan2(t_3, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	} else {
		tmp = atan2(((t_1 + (lambda1 * (cos(-lambda2) + (-0.5 * (lambda1 * t_1))))) * cos(phi2)), (-1.0 * t_2));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(-lambda2)
    t_2 = cos((lambda1 - lambda2)) * sin(phi1)
    t_3 = sin((lambda1 - lambda2)) * cos(phi2)
    if (lambda2 <= (-5.5d+66)) then
        tmp = atan2(t_3, (t_0 - t_2))
    else if (lambda2 <= 5.8d+46) then
        tmp = atan2(t_3, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))))
    else
        tmp = atan2(((t_1 + (lambda1 * (cos(-lambda2) + ((-0.5d0) * (lambda1 * t_1))))) * cos(phi2)), ((-1.0d0) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(-lambda2);
	double t_2 = Math.cos((lambda1 - lambda2)) * Math.sin(phi1);
	double t_3 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double tmp;
	if (lambda2 <= -5.5e+66) {
		tmp = Math.atan2(t_3, (t_0 - t_2));
	} else if (lambda2 <= 5.8e+46) {
		tmp = Math.atan2(t_3, (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else {
		tmp = Math.atan2(((t_1 + (lambda1 * (Math.cos(-lambda2) + (-0.5 * (lambda1 * t_1))))) * Math.cos(phi2)), (-1.0 * t_2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin(-lambda2)
	t_2 = math.cos((lambda1 - lambda2)) * math.sin(phi1)
	t_3 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	tmp = 0
	if lambda2 <= -5.5e+66:
		tmp = math.atan2(t_3, (t_0 - t_2))
	elif lambda2 <= 5.8e+46:
		tmp = math.atan2(t_3, (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	else:
		tmp = math.atan2(((t_1 + (lambda1 * (math.cos(-lambda2) + (-0.5 * (lambda1 * t_1))))) * math.cos(phi2)), (-1.0 * t_2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(-lambda2))
	t_2 = Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))
	t_3 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -5.5e+66)
		tmp = atan(t_3, Float64(t_0 - t_2));
	elseif (lambda2 <= 5.8e+46)
		tmp = atan(t_3, Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda1))));
	else
		tmp = atan(Float64(Float64(t_1 + Float64(lambda1 * Float64(cos(Float64(-lambda2)) + Float64(-0.5 * Float64(lambda1 * t_1))))) * cos(phi2)), Float64(-1.0 * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin(-lambda2);
	t_2 = cos((lambda1 - lambda2)) * sin(phi1);
	t_3 = sin((lambda1 - lambda2)) * cos(phi2);
	tmp = 0.0;
	if (lambda2 <= -5.5e+66)
		tmp = atan2(t_3, (t_0 - t_2));
	elseif (lambda2 <= 5.8e+46)
		tmp = atan2(t_3, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	else
		tmp = atan2(((t_1 + (lambda1 * (cos(-lambda2) + (-0.5 * (lambda1 * t_1))))) * cos(phi2)), (-1.0 * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[(-lambda2)], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -5.5e+66], N[ArcTan[t$95$3 / N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 5.8e+46], N[ArcTan[t$95$3 / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$1 + N[(lambda1 * N[(N[Cos[(-lambda2)], $MachinePrecision] + N[(-0.5 * N[(lambda1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -5.5e66

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. lower-sin.f64 66.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   if -5.5e66 < lambda2 < 5.8000000000000004e46

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]
   3. Step-by-step derivation

      1. lower-cos.f64 69.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   4. Applied rewrites 69.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}} \]

   if 5.8000000000000004e46 < lambda2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)}\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1} \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)}\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\frac{-1}{2}} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \color{blue}{\left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      11. lower-neg.f64 39.0%

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + -0.5 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 39.0%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + -0.5 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)} \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -8.8 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1
        (atan2
         (* (sin lambda1) (cos phi2))
         (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))))
  (if (<= lambda1 -8.8e-48)
    t_1
    (if (<= lambda1 2.7e+115)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (- t_0 (* (cos (- lambda1 lambda2)) (sin phi1))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	double tmp;
	if (lambda1 <= -8.8e-48) {
		tmp = t_1;
	} else if (lambda1 <= 2.7e+115) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = t_1;
	}
	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 = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    if (lambda1 <= (-8.8d-48)) then
        tmp = t_1
    else if (lambda1 <= 2.7d+115) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos((lambda1 - lambda2)) * sin(phi1))))
    else
        tmp = t_1
    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.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	double tmp;
	if (lambda1 <= -8.8e-48) {
		tmp = t_1;
	} else if (lambda1 <= 2.7e+115) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (Math.cos((lambda1 - lambda2)) * Math.sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	tmp = 0
	if lambda1 <= -8.8e-48:
		tmp = t_1
	elif lambda1 <= 2.7e+115:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (math.cos((lambda1 - lambda2)) * math.sin(phi1))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))))
	tmp = 0.0
	if (lambda1 <= -8.8e-48)
		tmp = t_1;
	elseif (lambda1 <= 2.7e+115)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	tmp = 0.0;
	if (lambda1 <= -8.8e-48)
		tmp = t_1;
	elseif (lambda1 <= 2.7e+115)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (cos((lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	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[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -8.8e-48], t$95$1, If[LessEqual[lambda1, 2.7e+115], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -8.8000000000000005e-48 or 2.7e115 < lambda1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot \sin \phi_1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]

   if -8.8000000000000005e-48 < lambda1 < 2.7e115

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. lower-sin.f64 66.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -40:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(0.5 \cdot {\lambda_1}^{2} - 1\right)\right) \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (cos phi1) (sin phi2)))
       (t_1 (cos (- lambda1 lambda2)))
       (t_2
        (atan2
         (* (sin (- lambda1 lambda2)) (cos phi2))
         (- t_0 (* t_1 (sin phi1))))))
  (if (<= (- lambda1 lambda2) -40.0)
    t_2
    (if (<= (- lambda1 lambda2) 5e-83)
      (atan2
       (*
        (+ lambda1 (* lambda2 (- (* 0.5 (pow lambda1 2.0)) 1.0)))
        (cos phi2))
       (- t_0 (* (* (sin phi1) (cos phi2)) t_1)))
      t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * sin(phi1))));
	double tmp;
	if ((lambda1 - lambda2) <= -40.0) {
		tmp = t_2;
	} else if ((lambda1 - lambda2) <= 5e-83) {
		tmp = atan2(((lambda1 + (lambda2 * ((0.5 * pow(lambda1, 2.0)) - 1.0))) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos((lambda1 - lambda2))
    t_2 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * sin(phi1))))
    if ((lambda1 - lambda2) <= (-40.0d0)) then
        tmp = t_2
    else if ((lambda1 - lambda2) <= 5d-83) then
        tmp = atan2(((lambda1 + (lambda2 * ((0.5d0 * (lambda1 ** 2.0d0)) - 1.0d0))) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * t_1)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos((lambda1 - lambda2));
	double t_2 = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (t_1 * Math.sin(phi1))));
	double tmp;
	if ((lambda1 - lambda2) <= -40.0) {
		tmp = t_2;
	} else if ((lambda1 - lambda2) <= 5e-83) {
		tmp = Math.atan2(((lambda1 + (lambda2 * ((0.5 * Math.pow(lambda1, 2.0)) - 1.0))) * Math.cos(phi2)), (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos((lambda1 - lambda2))
	t_2 = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (t_1 * math.sin(phi1))))
	tmp = 0
	if (lambda1 - lambda2) <= -40.0:
		tmp = t_2
	elif (lambda1 - lambda2) <= 5e-83:
		tmp = math.atan2(((lambda1 + (lambda2 * ((0.5 * math.pow(lambda1, 2.0)) - 1.0))) * math.cos(phi2)), (t_0 - ((math.sin(phi1) * math.cos(phi2)) * t_1)))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * sin(phi1))))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -40.0)
		tmp = t_2;
	elseif (Float64(lambda1 - lambda2) <= 5e-83)
		tmp = atan(Float64(Float64(lambda1 + Float64(lambda2 * Float64(Float64(0.5 * (lambda1 ^ 2.0)) - 1.0))) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos((lambda1 - lambda2));
	t_2 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * sin(phi1))));
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -40.0)
		tmp = t_2;
	elseif ((lambda1 - lambda2) <= 5e-83)
		tmp = atan2(((lambda1 + (lambda2 * ((0.5 * (lambda1 ^ 2.0)) - 1.0))) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * t_1)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -40.0], t$95$2, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 5e-83], N[ArcTan[N[(N[(lambda1 + N[(lambda2 * N[(N[(0.5 * N[Power[lambda1, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -40 or 4.9999999999999998e-83 < (-.f64 lambda1 lambda2)

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. lower-sin.f64 66.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   if -40 < (-.f64 lambda1 lambda2) < 4.9999999999999998e-83

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\lambda_1} \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \color{blue}{\frac{-1}{2}} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \color{blue}{\left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-neg.f64 59.6%

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + -0.5 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 59.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \left(\cos \left(-\lambda_2\right) + -0.5 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-pow.f64 39.4%

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(0.5 \cdot {\lambda_1}^{2} - 1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 39.4%

      \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(0.5 \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 66.0% accurate, 1.2× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (-
  (* (cos phi1) (sin phi2))
  (* (cos (- lambda1 lambda2)) (sin phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos((lambda1 - lambda2)) * Math.sin(phi1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (math.cos((lambda1 - lambda2)) * math.sin(phi1))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

   3. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   4. lower-sin.f64 66.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

4. Applied rewrites 66.0%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
5. Add Preprocessing

Alternative 25: 65.7% accurate, 1.2× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

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 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Step-by-step derivation

   1. lower-sin.f64 65.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

4. Applied rewrites 65.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
5. Add Preprocessing

Alternative 26: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -2.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 0.0023:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2)))
       (t_1 (atan2 t_0 (* (sin phi1) (- (cos (- lambda2 lambda1)))))))
  (if (<= phi1 -2.4)
    t_1
    (if (<= phi1 0.0023)
      (atan2
       t_0
       (+
        (sin phi2)
        (* -1.0 (* phi1 (* (cos phi2) (cos (- lambda1 lambda2)))))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_1 = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	double tmp;
	if (phi1 <= -2.4) {
		tmp = t_1;
	} else if (phi1 <= 0.0023) {
		tmp = atan2(t_0, (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * cos((lambda1 - lambda2)))))));
	} else {
		tmp = t_1;
	}
	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 = sin((lambda1 - lambda2)) * cos(phi2)
    t_1 = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))))
    if (phi1 <= (-2.4d0)) then
        tmp = t_1
    else if (phi1 <= 0.0023d0) then
        tmp = atan2(t_0, (sin(phi2) + ((-1.0d0) * (phi1 * (cos(phi2) * cos((lambda1 - lambda2)))))))
    else
        tmp = t_1
    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 t_1 = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	double tmp;
	if (phi1 <= -2.4) {
		tmp = t_1;
	} else if (phi1 <= 0.0023) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) + (-1.0 * (phi1 * (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	t_1 = math.atan2(t_0, (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	tmp = 0
	if phi1 <= -2.4:
		tmp = t_1
	elif phi1 <= 0.0023:
		tmp = math.atan2(t_0, (math.sin(phi2) + (-1.0 * (phi1 * (math.cos(phi2) * math.cos((lambda1 - lambda2)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_1 = atan(t_0, Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))))
	tmp = 0.0
	if (phi1 <= -2.4)
		tmp = t_1;
	elseif (phi1 <= 0.0023)
		tmp = atan(t_0, Float64(sin(phi2) + Float64(-1.0 * Float64(phi1 * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	t_1 = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	tmp = 0.0;
	if (phi1 <= -2.4)
		tmp = t_1;
	elseif (phi1 <= 0.0023)
		tmp = atan2(t_0, (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * cos((lambda1 - lambda2)))))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.4], t$95$1, If[LessEqual[phi1, 0.0023], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] + N[(-1.0 * N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.3999999999999999 or 0.0023 < phi1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. lower-neg.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      9. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

      12. sub-negate-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)} \]

      13. lower--.f64 48.7%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   if -2.3999999999999999 < phi1 < 0.0023

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + \color{blue}{-1} \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \color{blue}{\left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      8. lower--.f64 47.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 54.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)}\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{+35}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - t\_0 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (cos (- lambda1 lambda2)))
       (t_1
        (atan2
         (* (sin lambda1) (cos phi2))
         (+ (sin phi2) (* -1.0 (* phi1 (* (cos phi2) t_0)))))))
  (if (<= phi2 -3.7e+42)
    t_1
    (if (<= phi2 9.5e+35)
      (atan2
       (* (sin (- lambda1 lambda2)) (cos phi2))
       (- (* phi2 (cos phi1)) (* t_0 (sin phi1))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * t_0)))));
	double tmp;
	if (phi2 <= -3.7e+42) {
		tmp = t_1;
	} else if (phi2 <= 9.5e+35) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((phi2 * cos(phi1)) - (t_0 * sin(phi1))));
	} else {
		tmp = t_1;
	}
	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((lambda1 - lambda2))
    t_1 = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) + ((-1.0d0) * (phi1 * (cos(phi2) * t_0)))))
    if (phi2 <= (-3.7d+42)) then
        tmp = t_1
    else if (phi2 <= 9.5d+35) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((phi2 * cos(phi1)) - (t_0 * sin(phi1))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (Math.sin(phi2) + (-1.0 * (phi1 * (Math.cos(phi2) * t_0)))));
	double tmp;
	if (phi2 <= -3.7e+42) {
		tmp = t_1;
	} else if (phi2 <= 9.5e+35) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((phi2 * Math.cos(phi1)) - (t_0 * Math.sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.atan2((math.sin(lambda1) * math.cos(phi2)), (math.sin(phi2) + (-1.0 * (phi1 * (math.cos(phi2) * t_0)))))
	tmp = 0
	if phi2 <= -3.7e+42:
		tmp = t_1
	elif phi2 <= 9.5e+35:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((phi2 * math.cos(phi1)) - (t_0 * math.sin(phi1))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(sin(lambda1) * cos(phi2)), Float64(sin(phi2) + Float64(-1.0 * Float64(phi1 * Float64(cos(phi2) * t_0)))))
	tmp = 0.0
	if (phi2 <= -3.7e+42)
		tmp = t_1;
	elseif (phi2 <= 9.5e+35)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(phi2 * cos(phi1)) - Float64(t_0 * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) + (-1.0 * (phi1 * (cos(phi2) * t_0)))));
	tmp = 0.0;
	if (phi2 <= -3.7e+42)
		tmp = t_1;
	elseif (phi2 <= 9.5e+35)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((phi2 * cos(phi1)) - (t_0 * sin(phi1))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] + N[(-1.0 * N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -3.7e+42], t$95$1, If[LessEqual[phi2, 9.5e+35], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.7e42 or 9.5000000000000006e35 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 47.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + \color{blue}{-1} \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \color{blue}{\left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      8. lower--.f64 30.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

   7. Applied rewrites 30.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   if -3.7e42 < phi2 < 9.5000000000000006e35

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      6. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      7. lower-sin.f64 48.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   10. Applied rewrites 48.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 48.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.21:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{t\_0}\\ \mathbf{elif}\;\lambda_1 \leq 2.8 \cdot 10^{-44}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* -1.0 (* (cos lambda1) (sin phi1))))
       (t_1 (sin (- lambda1 lambda2)))
       (t_2 (* t_1 (cos phi2))))
  (if (<= lambda1 -0.21)
    (atan2 (* t_1 (+ 1.0 (* -0.5 (pow phi2 2.0)))) t_0)
    (if (<= lambda1 2.8e-44)
      (atan2 t_2 (* -1.0 (* (cos lambda2) (sin phi1))))
      (atan2 t_2 t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -1.0 * (cos(lambda1) * sin(phi1));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = t_1 * cos(phi2);
	double tmp;
	if (lambda1 <= -0.21) {
		tmp = atan2((t_1 * (1.0 + (-0.5 * pow(phi2, 2.0)))), t_0);
	} else if (lambda1 <= 2.8e-44) {
		tmp = atan2(t_2, (-1.0 * (cos(lambda2) * sin(phi1))));
	} else {
		tmp = atan2(t_2, t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) * (cos(lambda1) * sin(phi1))
    t_1 = sin((lambda1 - lambda2))
    t_2 = t_1 * cos(phi2)
    if (lambda1 <= (-0.21d0)) then
        tmp = atan2((t_1 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), t_0)
    else if (lambda1 <= 2.8d-44) then
        tmp = atan2(t_2, ((-1.0d0) * (cos(lambda2) * sin(phi1))))
    else
        tmp = atan2(t_2, t_0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -1.0 * (Math.cos(lambda1) * Math.sin(phi1));
	double t_1 = Math.sin((lambda1 - lambda2));
	double t_2 = t_1 * Math.cos(phi2);
	double tmp;
	if (lambda1 <= -0.21) {
		tmp = Math.atan2((t_1 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), t_0);
	} else if (lambda1 <= 2.8e-44) {
		tmp = Math.atan2(t_2, (-1.0 * (Math.cos(lambda2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_2, t_0);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = -1.0 * (math.cos(lambda1) * math.sin(phi1))
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = t_1 * math.cos(phi2)
	tmp = 0
	if lambda1 <= -0.21:
		tmp = math.atan2((t_1 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), t_0)
	elif lambda1 <= 2.8e-44:
		tmp = math.atan2(t_2, (-1.0 * (math.cos(lambda2) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_2, t_0)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-1.0 * Float64(cos(lambda1) * sin(phi1)))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (lambda1 <= -0.21)
		tmp = atan(Float64(t_1 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), t_0);
	elseif (lambda1 <= 2.8e-44)
		tmp = atan(t_2, Float64(-1.0 * Float64(cos(lambda2) * sin(phi1))));
	else
		tmp = atan(t_2, t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = -1.0 * (cos(lambda1) * sin(phi1));
	t_1 = sin((lambda1 - lambda2));
	t_2 = t_1 * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.21)
		tmp = atan2((t_1 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), t_0);
	elseif (lambda1 <= 2.8e-44)
		tmp = atan2(t_2, (-1.0 * (cos(lambda2) * sin(phi1))));
	else
		tmp = atan2(t_2, t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-1.0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.21], N[ArcTan[N[(t$95$1 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], If[LessEqual[lambda1, 2.8e-44], N[ArcTan[t$95$2 / N[(-1.0 * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / t$95$0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.20999999999999999

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower-pow.f64 28.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   10. Applied rewrites 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

       3. lower-sin.f64 39.3%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

   13. Applied rewrites 39.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]

   if -0.20999999999999999 < lambda1 < 2.8e-44

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \phi_1\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \phi_1\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \phi_1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \phi_1\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \sin \phi_1\right)} \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1\right)} \]

      12. lower-*.f64 48.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \sin \phi_1\right)} \]

   6. Applied rewrites 48.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_2 \cdot \color{blue}{\sin \phi_1}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)} \]

      3. lower-sin.f64 43.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)} \]

   9. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_2 \cdot \color{blue}{\sin \phi_1}\right)} \]

   if 2.8e-44 < lambda1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

      3. lower-sin.f64 43.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 48.4% accurate, 1.7× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (* (sin phi1) (- (cos (- lambda2 lambda1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(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)), (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.sin(phi1) * -Math.cos((lambda2 - lambda1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(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(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(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[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   7. lower-neg.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   8. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   9. cos-neg-rev N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \]

   12. sub-negate-rev N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   13. lower--.f64 48.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \color{blue}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]
7. Add Preprocessing

Alternative 30: 47.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\phi_2 \leq -28000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 1.45 \cdot 10^{-32}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (sin (- lambda1 lambda2)))
       (t_1
        (atan2
         (* t_0 (cos phi2))
         (* -1.0 (* (cos lambda1) (sin phi1))))))
  (if (<= phi2 -28000000000000.0)
    t_1
    (if (<= phi2 1.45e-32)
      (atan2
       (* t_0 (+ 1.0 (* -0.5 (pow phi2 2.0))))
       (* -1.0 (* (cos (- lambda1 lambda2)) (sin phi1))))
      t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), (-1.0 * (cos(lambda1) * sin(phi1))));
	double tmp;
	if (phi2 <= -28000000000000.0) {
		tmp = t_1;
	} else if (phi2 <= 1.45e-32) {
		tmp = atan2((t_0 * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = t_1;
	}
	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 = sin((lambda1 - lambda2))
    t_1 = atan2((t_0 * cos(phi2)), ((-1.0d0) * (cos(lambda1) * sin(phi1))))
    if (phi2 <= (-28000000000000.0d0)) then
        tmp = t_1
    else if (phi2 <= 1.45d-32) then
        tmp = atan2((t_0 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (cos((lambda1 - lambda2)) * sin(phi1))))
    else
        tmp = t_1
    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 t_1 = Math.atan2((t_0 * Math.cos(phi2)), (-1.0 * (Math.cos(lambda1) * Math.sin(phi1))));
	double tmp;
	if (phi2 <= -28000000000000.0) {
		tmp = t_1;
	} else if (phi2 <= 1.45e-32) {
		tmp = Math.atan2((t_0 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (Math.cos((lambda1 - lambda2)) * Math.sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.atan2((t_0 * math.cos(phi2)), (-1.0 * (math.cos(lambda1) * math.sin(phi1))))
	tmp = 0
	if phi2 <= -28000000000000.0:
		tmp = t_1
	elif phi2 <= 1.45e-32:
		tmp = math.atan2((t_0 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (math.cos((lambda1 - lambda2)) * math.sin(phi1))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), Float64(-1.0 * Float64(cos(lambda1) * sin(phi1))))
	tmp = 0.0
	if (phi2 <= -28000000000000.0)
		tmp = t_1;
	elseif (phi2 <= 1.45e-32)
		tmp = atan(Float64(t_0 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((t_0 * cos(phi2)), (-1.0 * (cos(lambda1) * sin(phi1))));
	tmp = 0.0;
	if (phi2 <= -28000000000000.0)
		tmp = t_1;
	elseif (phi2 <= 1.45e-32)
		tmp = atan2((t_0 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (cos((lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -28000000000000.0], t$95$1, If[LessEqual[phi2, 1.45e-32], N[ArcTan[N[(t$95$0 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.8e13 or 1.45e-32 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

      3. lower-sin.f64 43.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]

   if -2.8e13 < phi2 < 1.45e-32

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. lower-pow.f64 44.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 44.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 47.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 3.6:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(t\_0 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(t\_0 \cdot \left(\phi_1 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_1}^{2}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (cos (- lambda1 lambda2)))
       (t_1 (sin (- lambda1 lambda2)))
       (t_2 (* t_1 (cos phi2))))
  (if (<= phi2 -5.5)
    (atan2 t_2 (* -1.0 (* phi1 (cos (- lambda2)))))
    (if (<= phi2 3.6)
      (atan2
       (* t_1 (+ 1.0 (* -0.5 (pow phi2 2.0))))
       (* -1.0 (* t_0 (sin phi1))))
      (atan2
       t_2
       (*
        -1.0
        (*
         t_0
         (*
          phi1
          (+ 1.0 (* -0.16666666666666666 (pow phi1 2.0)))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = t_1 * cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = atan2(t_2, (-1.0 * (phi1 * cos(-lambda2))));
	} else if (phi2 <= 3.6) {
		tmp = atan2((t_1 * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (t_0 * sin(phi1))));
	} else {
		tmp = atan2(t_2, (-1.0 * (t_0 * (phi1 * (1.0 + (-0.16666666666666666 * pow(phi1, 2.0)))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin((lambda1 - lambda2))
    t_2 = t_1 * cos(phi2)
    if (phi2 <= (-5.5d0)) then
        tmp = atan2(t_2, ((-1.0d0) * (phi1 * cos(-lambda2))))
    else if (phi2 <= 3.6d0) then
        tmp = atan2((t_1 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (t_0 * sin(phi1))))
    else
        tmp = atan2(t_2, ((-1.0d0) * (t_0 * (phi1 * (1.0d0 + ((-0.16666666666666666d0) * (phi1 ** 2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.sin((lambda1 - lambda2));
	double t_2 = t_1 * Math.cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = Math.atan2(t_2, (-1.0 * (phi1 * Math.cos(-lambda2))));
	} else if (phi2 <= 3.6) {
		tmp = Math.atan2((t_1 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (t_0 * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_2, (-1.0 * (t_0 * (phi1 * (1.0 + (-0.16666666666666666 * Math.pow(phi1, 2.0)))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = t_1 * math.cos(phi2)
	tmp = 0
	if phi2 <= -5.5:
		tmp = math.atan2(t_2, (-1.0 * (phi1 * math.cos(-lambda2))))
	elif phi2 <= 3.6:
		tmp = math.atan2((t_1 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (t_0 * math.sin(phi1))))
	else:
		tmp = math.atan2(t_2, (-1.0 * (t_0 * (phi1 * (1.0 + (-0.16666666666666666 * math.pow(phi1, 2.0)))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -5.5)
		tmp = atan(t_2, Float64(-1.0 * Float64(phi1 * cos(Float64(-lambda2)))));
	elseif (phi2 <= 3.6)
		tmp = atan(Float64(t_1 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(t_0 * sin(phi1))));
	else
		tmp = atan(t_2, Float64(-1.0 * Float64(t_0 * Float64(phi1 * Float64(1.0 + Float64(-0.16666666666666666 * (phi1 ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin((lambda1 - lambda2));
	t_2 = t_1 * cos(phi2);
	tmp = 0.0;
	if (phi2 <= -5.5)
		tmp = atan2(t_2, (-1.0 * (phi1 * cos(-lambda2))));
	elseif (phi2 <= 3.6)
		tmp = atan2((t_1 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (t_0 * sin(phi1))));
	else
		tmp = atan2(t_2, (-1.0 * (t_0 * (phi1 * (1.0 + (-0.16666666666666666 * (phi1 ^ 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.5], N[ArcTan[t$95$2 / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 3.6], N[ArcTan[N[(t$95$1 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(-1.0 * N[(t$95$0 * N[(phi1 * N[(1.0 + N[(-0.16666666666666666 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

      2. lower-neg.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   10. Applied rewrites 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   if -5.5 < phi2 < 3.6000000000000001

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. lower-pow.f64 44.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 44.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   if 3.6000000000000001 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_1}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\phi_1}^{2}}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{\phi_1}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \left(1 + \frac{-1}{6} \cdot {\phi_1}^{\color{blue}{2}}\right)\right)\right)} \]

      4. lower-pow.f64 31.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_1}^{2}\right)\right)\right)} \]

   7. Applied rewrites 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_1 \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\phi_1}^{2}\right)}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 32: 47.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := t\_0 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* t_0 (cos phi2))))
  (if (<= phi2 -5.5)
    (atan2 t_1 (* -1.0 (* phi1 (cos (- lambda2)))))
    (if (<= phi2 8.8e+15)
      (atan2
       (* t_0 (+ 1.0 (* -0.5 (pow phi2 2.0))))
       (* -1.0 (* (cos (- lambda1 lambda2)) (sin phi1))))
      (atan2
       t_1
       (* -1.0 (* (+ 1.0 (* lambda1 lambda2)) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = t_0 * cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = atan2(t_1, (-1.0 * (phi1 * cos(-lambda2))));
	} else if (phi2 <= 8.8e+15) {
		tmp = atan2((t_0 * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	}
	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 = sin((lambda1 - lambda2))
    t_1 = t_0 * cos(phi2)
    if (phi2 <= (-5.5d0)) then
        tmp = atan2(t_1, ((-1.0d0) * (phi1 * cos(-lambda2))))
    else if (phi2 <= 8.8d+15) then
        tmp = atan2((t_0 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (cos((lambda1 - lambda2)) * sin(phi1))))
    else
        tmp = atan2(t_1, ((-1.0d0) * ((1.0d0 + (lambda1 * lambda2)) * sin(phi1))))
    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 t_1 = t_0 * Math.cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = Math.atan2(t_1, (-1.0 * (phi1 * Math.cos(-lambda2))));
	} else if (phi2 <= 8.8e+15) {
		tmp = Math.atan2((t_0 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (Math.cos((lambda1 - lambda2)) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = t_0 * math.cos(phi2)
	tmp = 0
	if phi2 <= -5.5:
		tmp = math.atan2(t_1, (-1.0 * (phi1 * math.cos(-lambda2))))
	elif phi2 <= 8.8e+15:
		tmp = math.atan2((t_0 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (math.cos((lambda1 - lambda2)) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(t_0 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -5.5)
		tmp = atan(t_1, Float64(-1.0 * Float64(phi1 * cos(Float64(-lambda2)))));
	elseif (phi2 <= 8.8e+15)
		tmp = atan(Float64(t_0 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = atan(t_1, Float64(-1.0 * Float64(Float64(1.0 + Float64(lambda1 * lambda2)) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = t_0 * cos(phi2);
	tmp = 0.0;
	if (phi2 <= -5.5)
		tmp = atan2(t_1, (-1.0 * (phi1 * cos(-lambda2))));
	elseif (phi2 <= 8.8e+15)
		tmp = atan2((t_0 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (cos((lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.5], N[ArcTan[t$95$1 / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 8.8e+15], N[ArcTan[N[(t$95$0 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(-1.0 * N[(N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

      2. lower-neg.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   10. Applied rewrites 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   if -5.5 < phi2 < 8.8e15

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      3. lower-pow.f64 44.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 44.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   if 8.8e15 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. lower-neg.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 33.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

   10. Applied rewrites 33.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 33: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-\lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(\phi_1 \cdot t\_0\right)}\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(t\_0 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (cos (- lambda2)))
       (t_1 (sin (- lambda1 lambda2)))
       (t_2 (* t_1 (cos phi2))))
  (if (<= phi2 -5.5)
    (atan2 t_2 (* -1.0 (* phi1 t_0)))
    (if (<= phi2 8.8e+15)
      (atan2
       (* t_1 (+ 1.0 (* -0.5 (pow phi2 2.0))))
       (* -1.0 (* t_0 (sin phi1))))
      (atan2
       t_2
       (* -1.0 (* (+ 1.0 (* lambda1 lambda2)) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(-lambda2);
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = t_1 * cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = atan2(t_2, (-1.0 * (phi1 * t_0)));
	} else if (phi2 <= 8.8e+15) {
		tmp = atan2((t_1 * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (t_0 * sin(phi1))));
	} else {
		tmp = atan2(t_2, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	}
	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(-lambda2)
    t_1 = sin((lambda1 - lambda2))
    t_2 = t_1 * cos(phi2)
    if (phi2 <= (-5.5d0)) then
        tmp = atan2(t_2, ((-1.0d0) * (phi1 * t_0)))
    else if (phi2 <= 8.8d+15) then
        tmp = atan2((t_1 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (t_0 * sin(phi1))))
    else
        tmp = atan2(t_2, ((-1.0d0) * ((1.0d0 + (lambda1 * lambda2)) * sin(phi1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(-lambda2);
	double t_1 = Math.sin((lambda1 - lambda2));
	double t_2 = t_1 * Math.cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = Math.atan2(t_2, (-1.0 * (phi1 * t_0)));
	} else if (phi2 <= 8.8e+15) {
		tmp = Math.atan2((t_1 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (t_0 * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_2, (-1.0 * ((1.0 + (lambda1 * lambda2)) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(-lambda2)
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = t_1 * math.cos(phi2)
	tmp = 0
	if phi2 <= -5.5:
		tmp = math.atan2(t_2, (-1.0 * (phi1 * t_0)))
	elif phi2 <= 8.8e+15:
		tmp = math.atan2((t_1 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (t_0 * math.sin(phi1))))
	else:
		tmp = math.atan2(t_2, (-1.0 * ((1.0 + (lambda1 * lambda2)) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -5.5)
		tmp = atan(t_2, Float64(-1.0 * Float64(phi1 * t_0)));
	elseif (phi2 <= 8.8e+15)
		tmp = atan(Float64(t_1 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(t_0 * sin(phi1))));
	else
		tmp = atan(t_2, Float64(-1.0 * Float64(Float64(1.0 + Float64(lambda1 * lambda2)) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(-lambda2);
	t_1 = sin((lambda1 - lambda2));
	t_2 = t_1 * cos(phi2);
	tmp = 0.0;
	if (phi2 <= -5.5)
		tmp = atan2(t_2, (-1.0 * (phi1 * t_0)));
	elseif (phi2 <= 8.8e+15)
		tmp = atan2((t_1 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (t_0 * sin(phi1))));
	else
		tmp = atan2(t_2, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[(-lambda2)], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.5], N[ArcTan[t$95$2 / N[(-1.0 * N[(phi1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 8.8e+15], N[ArcTan[N[(t$95$1 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(-1.0 * N[(N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

      2. lower-neg.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   10. Applied rewrites 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   if -5.5 < phi2 < 8.8e15

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower-pow.f64 28.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   10. Applied rewrites 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\sin \phi_1}\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)} \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right)} \]

       3. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)} \]

       4. lower-sin.f64 39.3%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)} \]

   13. Applied rewrites 39.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \left(-\lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   if 8.8e15 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. lower-neg.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 33.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

   10. Applied rewrites 33.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 34: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := t\_0 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* t_0 (cos phi2))))
  (if (<= phi2 -5.5)
    (atan2 t_1 (* -1.0 (* phi1 (cos (- lambda2)))))
    (if (<= phi2 8.8e+15)
      (atan2
       (* t_0 (+ 1.0 (* -0.5 (pow phi2 2.0))))
       (* -1.0 (* (cos lambda1) (sin phi1))))
      (atan2
       t_1
       (* -1.0 (* (+ 1.0 (* lambda1 lambda2)) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = t_0 * cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = atan2(t_1, (-1.0 * (phi1 * cos(-lambda2))));
	} else if (phi2 <= 8.8e+15) {
		tmp = atan2((t_0 * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (cos(lambda1) * sin(phi1))));
	} else {
		tmp = atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	}
	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 = sin((lambda1 - lambda2))
    t_1 = t_0 * cos(phi2)
    if (phi2 <= (-5.5d0)) then
        tmp = atan2(t_1, ((-1.0d0) * (phi1 * cos(-lambda2))))
    else if (phi2 <= 8.8d+15) then
        tmp = atan2((t_0 * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (cos(lambda1) * sin(phi1))))
    else
        tmp = atan2(t_1, ((-1.0d0) * ((1.0d0 + (lambda1 * lambda2)) * sin(phi1))))
    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 t_1 = t_0 * Math.cos(phi2);
	double tmp;
	if (phi2 <= -5.5) {
		tmp = Math.atan2(t_1, (-1.0 * (phi1 * Math.cos(-lambda2))));
	} else if (phi2 <= 8.8e+15) {
		tmp = Math.atan2((t_0 * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (Math.cos(lambda1) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = t_0 * math.cos(phi2)
	tmp = 0
	if phi2 <= -5.5:
		tmp = math.atan2(t_1, (-1.0 * (phi1 * math.cos(-lambda2))))
	elif phi2 <= 8.8e+15:
		tmp = math.atan2((t_0 * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (math.cos(lambda1) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(t_0 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -5.5)
		tmp = atan(t_1, Float64(-1.0 * Float64(phi1 * cos(Float64(-lambda2)))));
	elseif (phi2 <= 8.8e+15)
		tmp = atan(Float64(t_0 * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(cos(lambda1) * sin(phi1))));
	else
		tmp = atan(t_1, Float64(-1.0 * Float64(Float64(1.0 + Float64(lambda1 * lambda2)) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = t_0 * cos(phi2);
	tmp = 0.0;
	if (phi2 <= -5.5)
		tmp = atan2(t_1, (-1.0 * (phi1 * cos(-lambda2))));
	elseif (phi2 <= 8.8e+15)
		tmp = atan2((t_0 * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (cos(lambda1) * sin(phi1))));
	else
		tmp = atan2(t_1, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.5], N[ArcTan[t$95$1 / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 8.8e+15], N[ArcTan[N[(t$95$0 * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(-1.0 * N[(N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

      2. lower-neg.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   10. Applied rewrites 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   if -5.5 < phi2 < 8.8e15

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower-pow.f64 28.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   10. Applied rewrites 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

       3. lower-sin.f64 39.3%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)} \]

   13. Applied rewrites 39.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{\sin \phi_1}\right)} \]

   if 8.8e15 < phi2

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. lower-neg.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 33.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

   10. Applied rewrites 33.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 35: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
  (if (<= phi1 -6.5e+14)
    (atan2 t_0 (* -1.0 (* (+ 1.0 (* lambda1 lambda2)) (sin phi1))))
    (atan2 t_0 (* -1.0 (* 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 (phi1 <= -6.5e+14) {
		tmp = atan2(t_0, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	} else {
		tmp = atan2(t_0, (-1.0 * (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 (phi1 <= (-6.5d+14)) then
        tmp = atan2(t_0, ((-1.0d0) * ((1.0d0 + (lambda1 * lambda2)) * sin(phi1))))
    else
        tmp = atan2(t_0, ((-1.0d0) * (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 (phi1 <= -6.5e+14) {
		tmp = Math.atan2(t_0, (-1.0 * ((1.0 + (lambda1 * lambda2)) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_0, (-1.0 * (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 phi1 <= -6.5e+14:
		tmp = math.atan2(t_0, (-1.0 * ((1.0 + (lambda1 * lambda2)) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_0, (-1.0 * (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 (phi1 <= -6.5e+14)
		tmp = atan(t_0, Float64(-1.0 * Float64(Float64(1.0 + Float64(lambda1 * lambda2)) * sin(phi1))));
	else
		tmp = atan(t_0, Float64(-1.0 * Float64(phi1 * cos(Float64(-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 (phi1 <= -6.5e+14)
		tmp = atan2(t_0, (-1.0 * ((1.0 + (lambda1 * lambda2)) * sin(phi1))));
	else
		tmp = atan2(t_0, (-1.0 * (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[phi1, -6.5e+14], N[ArcTan[t$95$0 / N[(-1.0 * N[(N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.5e14

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. lower-neg.f64 40.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

   7. Applied rewrites 40.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(\cos \left(-\lambda_2\right) + -1 \cdot \left(\lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-*.f64 33.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

   10. Applied rewrites 33.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\left(1 + \lambda_1 \cdot \lambda_2\right) \cdot \sin \phi_1\right)} \]

   if -6.5e14 < phi1

   1. Initial program 79.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      5. lower-sin.f64 48.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      3. lower--.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

      2. lower-neg.f64 31.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

   10. Applied rewrites 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 36: 31.6% accurate, 2.1× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (* -1.0 (* phi1 (cos (- lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), (-1.0 * (phi1 * cos(-lambda2))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((-1.0d0) * (phi1 * cos(-lambda2))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (-1.0 * (phi1 * Math.cos(-lambda2))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (-1.0 * (phi1 * math.cos(-lambda2))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(-1.0 * Float64(phi1 * cos(Float64(-lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (-1.0 * (phi1 * cos(-lambda2))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower--.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

7. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

8. Taylor expanded in lambda1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
9. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

   2. lower-neg.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

10. Applied rewrites 31.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]
11. Add Preprocessing

Alternative 37: 31.6% accurate, 2.1× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (- (* (cos (- lambda2 lambda1)) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), -(cos((lambda2 - lambda1)) * phi1));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), -(cos((lambda2 - lambda1)) * phi1))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), -(Math.cos((lambda2 - lambda1)) * phi1));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), -(math.cos((lambda2 - lambda1)) * phi1))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * phi1)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), -(cos((lambda2 - lambda1)) * 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[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower--.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

7. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   2. mul-1-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower-neg.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1} \]

   6. lower-*.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1} \]

   7. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1} \]

   8. cos-neg-rev N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \phi_1} \]

   9. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \phi_1} \]

   10. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \phi_1} \]

   11. sub-negate-rev N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   12. lower--.f64 31.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

9. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
10. Add Preprocessing

Alternative 38: 31.5% accurate, 2.1× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (cos phi2))
 (* -1.0 (* phi1 (cos lambda1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), (-1.0 * (phi1 * cos(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)), ((-1.0d0) * (phi1 * cos(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)), (-1.0 * (phi1 * Math.cos(lambda1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (-1.0 * (phi1 * math.cos(lambda1))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(-1.0 * Float64(phi1 * cos(lambda1))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (-1.0 * (phi1 * cos(lambda1))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(phi1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower--.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

7. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)} \]

   2. lower-cos.f64 31.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)} \]

10. Applied rewrites 31.5%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)} \]
11. Add Preprocessing

Alternative 39: 28.6% accurate, 2.3× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (sin (- lambda1 lambda2)) (+ 1.0 (* -0.5 (pow phi2 2.0))))
 (* -1.0 (* phi1 (cos (- lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * (1.0 + (-0.5 * pow(phi2, 2.0)))), (-1.0 * (phi1 * cos(-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)) * (1.0d0 + ((-0.5d0) * (phi2 ** 2.0d0)))), ((-1.0d0) * (phi1 * cos(-lambda2))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * (1.0 + (-0.5 * Math.pow(phi2, 2.0)))), (-1.0 * (phi1 * Math.cos(-lambda2))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * (1.0 + (-0.5 * math.pow(phi2, 2.0)))), (-1.0 * (phi1 * math.cos(-lambda2))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * Float64(1.0 + Float64(-0.5 * (phi2 ^ 2.0)))), Float64(-1.0 * Float64(phi1 * cos(Float64(-lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * (1.0 + (-0.5 * (phi2 ^ 2.0)))), (-1.0 * (phi1 * cos(-lambda2))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(phi1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower--.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

7. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

8. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower-pow.f64 28.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

10. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

11. Taylor expanded in lambda1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]
12. Step-by-step derivation

    1. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \]

    2. lower-neg.f64 28.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]

13. Applied rewrites 28.5%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{2}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(-\lambda_2\right)\right)} \]
14. Add Preprocessing

Alternative 40: 28.5% accurate, 2.6× speedup

Math

\[\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5 \cdot \phi_2, \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
  :precision binary64
  (atan2
 (* (fma (* -0.5 phi2) phi2 1.0) (sin (- lambda1 lambda2)))
 (- (* (cos (- lambda2 lambda1)) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma((-0.5 * phi2), phi2, 1.0) * sin((lambda1 - lambda2))), -(cos((lambda2 - lambda1)) * phi1));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(Float64(-0.5 * phi2), phi2, 1.0) * sin(Float64(lambda1 - lambda2))), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * phi1)))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(N[(-0.5 * phi2), $MachinePrecision] * phi2 + 1.0), $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 79.2%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   5. lower-sin.f64 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower--.f64 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

7. Applied rewrites 31.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

8. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   3. lower-pow.f64 28.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

10. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

12. Applied rewrites 28.6%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5 \cdot \phi_2, \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))))