Bearing on a great circle

Percentage Accurate: 79.1% → 99.7%

Time: 18.6s

Alternatives: 42

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. negate-sub 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. sin-sum-rev N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. sin-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 89.6

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

3. Applied rewrites 89.6%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. negate-sub2 N/A

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

   4. cos-neg-rev N/A

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

   5. cos-diff N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. mul-1-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-neg-rev N/A

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

   11. mul-1-neg N/A

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

   12. sin-sum N/A

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

   13. mul-1-neg N/A

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

   14. negate-sub N/A

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

   15. sin-diff N/A

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

   16. *-commutative N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. lift-sin.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

Alternative 2: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := 1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0024:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 0.001:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot t\_2}{t\_0 - \left(\sin \phi_1 \cdot t\_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (- t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2 (- 1.0 (* 0.5 (* phi2 phi2)))))
   (if (<= phi2 -0.0024)
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      t_1)
     (if (<= phi2 0.001)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         t_2)
        (-
         t_0
         (*
          (* (sin phi1) t_2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
         (cos phi2))
        t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = t_0 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = 1.0 - (0.5 * (phi2 * phi2));
	double tmp;
	if (phi2 <= -0.0024) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), t_1);
	} else if (phi2 <= 0.001) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * t_2), (t_0 - ((sin(phi1) * t_2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(phi2 * phi2)))
	tmp = 0.0
	if (phi2 <= -0.0024)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), t_1);
	elseif (phi2 <= 0.001)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * t_2), Float64(t_0 - Float64(Float64(sin(phi1) * t_2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), t_1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.0024], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[phi2, 0.001], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.00239999999999999979

   1. Initial program 76.2%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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. cos-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 90.0

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

   3. Applied rewrites 90.0%

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

   if -0.00239999999999999979 < phi2 < 1e-3

   1. Initial program 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 89.4

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

   3. Applied rewrites 89.4%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-neg-rev N/A

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

      11. mul-1-neg N/A

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

      12. sin-sum N/A

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

      13. mul-1-neg N/A

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

      14. negate-sub N/A

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

      15. sin-diff N/A

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

      16. *-commutative N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-sin.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {\phi_2}^{2}}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {\phi_2}^{2}}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 99.9

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - 0.5 \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   10. Applied rewrites 99.9%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \color{blue}{\left(1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
   12. Step-by-step derivation

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

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {\phi_2}^{2}}\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {\phi_2}^{2}}\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 99.9

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(1 - 0.5 \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   13. Applied rewrites 99.9%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   if 1e-3 < phi2

   1. Initial program 77.1%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 89.7

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

   3. Applied rewrites 89.7%

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

Alternative 3: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq -0.0024:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_2}\\ \mathbf{elif}\;\phi_2 \leq 0.001:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot t\_3}{t\_1 - \left(\sin \phi_1 \cdot t\_3\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (- t_1 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_3 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 -0.0024)
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      t_2)
     (if (<= phi2 0.001)
       (atan2
        (* t_0 t_3)
        (-
         t_1
         (*
          (* (sin phi1) t_3)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
       (atan2 (* t_0 (cos phi2)) t_2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = t_1 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_3 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= -0.0024) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), t_2);
	} else if (phi2 <= 0.001) {
		tmp = atan2((t_0 * t_3), (t_1 - ((sin(phi1) * t_3) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = atan2((t_0 * cos(phi2)), t_2);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(t_1 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_3 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= -0.0024)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), t_2);
	elseif (phi2 <= 0.001)
		tmp = atan(Float64(t_0 * t_3), Float64(t_1 - Float64(Float64(sin(phi1) * t_3) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = atan(Float64(t_0 * cos(phi2)), t_2);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, -0.0024], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision], If[LessEqual[phi2, 0.001], N[ArcTan[N[(t$95$0 * t$95$3), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.00239999999999999979

   1. Initial program 76.2%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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. cos-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 90.0

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

   3. Applied rewrites 90.0%

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

   if -0.00239999999999999979 < phi2 < 1e-3

   1. Initial program 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 89.4

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

   3. Applied rewrites 89.4%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. lower-*.f64 99.9

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

       5. lower-*.f64 99.9

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   11. Applied rewrites 99.9%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   if 1e-3 < phi2

   1. Initial program 77.1%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 89.7

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

   3. Applied rewrites 89.7%

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

Alternative 4: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 8.4 \cdot 10^{+40}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (- (sin lambda2))))
           (cos phi2))
          (- t_0 (* t_1 (cos lambda2))))))
   (if (<= lambda2 -2.1e-11)
     t_2
     (if (<= lambda2 8.4e+40)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (-
         t_0
         (*
          t_1
          (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 = sin(phi1) * cos(phi2);
	double t_2 = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	double tmp;
	if (lambda2 <= -2.1e-11) {
		tmp = t_2;
	} else if (lambda2 <= 8.4e+40) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * 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(sin(phi1) * cos(phi2))
	t_2 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))))
	tmp = 0.0
	if (lambda2 <= -2.1e-11)
		tmp = t_2;
	elseif (lambda2 <= 8.4e+40)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * 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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -2.1e-11], t$95$2, If[LessEqual[lambda2, 8.4e+40], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * 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 lambda2 < -2.0999999999999999e-11 or 8.4000000000000004e40 < lambda2

   1. Initial program 60.1%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 80.4

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

   3. Applied rewrites 80.4%

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

   4. Taylor expanded in lambda1 around 0

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

      1. negate-sub2 N/A

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

      2. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-cos.f64 80.2

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

   6. Applied rewrites 80.2%

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

   if -2.0999999999999999e-11 < lambda2 < 8.4000000000000004e40

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      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. cos-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 96.8

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

   3. Applied rewrites 96.8%

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

Alternative 5: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \lambda_2, \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (- (sin lambda2))))
           (cos phi2))
          (- t_0 (* t_1 (cos lambda2))))))
   (if (<= lambda2 -2.1e-11)
     t_2
     (if (<= lambda2 6e-7)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* t_1 (fma (sin lambda1) lambda2 (cos lambda1)))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double t_2 = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	double tmp;
	if (lambda2 <= -2.1e-11) {
		tmp = t_2;
	} else if (lambda2 <= 6e-7) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * fma(sin(lambda1), lambda2, cos(lambda1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))))
	tmp = 0.0
	if (lambda2 <= -2.1e-11)
		tmp = t_2;
	elseif (lambda2 <= 6e-7)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * fma(sin(lambda1), lambda2, cos(lambda1)))));
	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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -2.1e-11], t$95$2, If[LessEqual[lambda2, 6e-7], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Sin[lambda1], $MachinePrecision] * lambda2 + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -2.0999999999999999e-11 or 5.9999999999999997e-7 < lambda2

   1. Initial program 60.0%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in lambda1 around 0

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

      1. negate-sub2 N/A

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

      2. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-cos.f64 80.0

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

   6. Applied rewrites 80.0%

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

   if -2.0999999999999999e-11 < lambda2 < 5.9999999999999997e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 6: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \lambda_2, \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (- t_0 (* t_1 (cos lambda2))))))
   (if (<= lambda2 -2.1e-11)
     t_2
     (if (<= lambda2 6e-7)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* t_1 (fma (sin lambda1) lambda2 (cos lambda1)))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double t_2 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	double tmp;
	if (lambda2 <= -2.1e-11) {
		tmp = t_2;
	} else if (lambda2 <= 6e-7) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * fma(sin(lambda1), lambda2, cos(lambda1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))))
	tmp = 0.0
	if (lambda2 <= -2.1e-11)
		tmp = t_2;
	elseif (lambda2 <= 6e-7)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * fma(sin(lambda1), lambda2, cos(lambda1)))));
	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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -2.1e-11], t$95$2, If[LessEqual[lambda2, 6e-7], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Sin[lambda1], $MachinePrecision] * lambda2 + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -2.0999999999999999e-11 or 5.9999999999999997e-7 < lambda2

   1. Initial program 60.0%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 80.3

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

   3. Applied rewrites 80.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-neg-rev N/A

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

      11. mul-1-neg N/A

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

      12. sin-sum N/A

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

      13. mul-1-neg N/A

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

      14. negate-sub N/A

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

      15. sin-diff N/A

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

      16. *-commutative N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-sin.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in lambda1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-cos.f64 80.0

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

   10. Applied rewrites 80.0%

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

   if -2.0999999999999999e-11 < lambda2 < 5.9999999999999997e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 7: 89.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. negate-sub 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. sin-sum-rev N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. sin-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 89.6

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

3. Applied rewrites 89.6%

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

Alternative 8: 88.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin 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) * cos(lambda2)) - (cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(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) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(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(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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. cos-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 89.6

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

3. Applied rewrites 89.6%

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

Alternative 9: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, 1, t\_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0}\\ \mathbf{if}\;\phi_1 \leq -20:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, t\_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), \phi_1 \cdot \phi_1, 1\right) \cdot \sin \phi_2 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))
        (t_1 (* (cos lambda1) (- (sin lambda2))))
        (t_2
         (atan2
          (* (fma (sin lambda1) 1.0 t_1) (cos phi2))
          (- (* (cos phi1) (sin phi2)) t_0))))
   (if (<= phi1 -20.0)
     t_2
     (if (<= phi1 19.0)
       (atan2
        (* (fma (sin lambda1) (cos lambda2) t_1) (cos phi2))
        (-
         (*
          (fma (fma (* phi1 phi1) 0.041666666666666664 -0.5) (* phi1 phi1) 1.0)
          (sin phi2))
         t_0))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2));
	double t_1 = cos(lambda1) * -sin(lambda2);
	double t_2 = atan2((fma(sin(lambda1), 1.0, t_1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - t_0));
	double tmp;
	if (phi1 <= -20.0) {
		tmp = t_2;
	} else if (phi1 <= 19.0) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), t_1) * cos(phi2)), ((fma(fma((phi1 * phi1), 0.041666666666666664, -0.5), (phi1 * phi1), 1.0) * sin(phi2)) - t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(lambda1) * Float64(-sin(lambda2)))
	t_2 = atan(Float64(fma(sin(lambda1), 1.0, t_1) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - t_0))
	tmp = 0.0
	if (phi1 <= -20.0)
		tmp = t_2;
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), t_1) * cos(phi2)), Float64(Float64(fma(fma(Float64(phi1 * phi1), 0.041666666666666664, -0.5), Float64(phi1 * phi1), 1.0) * sin(phi2)) - t_0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * 1.0 + t$95$1), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -20.0], t$95$2, If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$1), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(phi1 * phi1), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -20 or 19 < phi1

   1. Initial program 76.7%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 79.6

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

   3. Applied rewrites 79.6%

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

   4. Taylor expanded in lambda2 around 0

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

   6. Applied rewrites 77.7%

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

   if -20 < phi1 < 19

   1. Initial program 81.4%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\left(1 + {\phi_1}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}\right)\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left({\phi_1}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(\left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}\right) \cdot {\phi_1}^{2} + 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}, \color{blue}{{\phi_1}^{2}}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. negate-sub N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{24} \cdot {\phi_1}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{\phi_1}}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_1}^{2} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\phi_1}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_1}^{2} \cdot \frac{1}{24} + \frac{-1}{2}, {\phi_1}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left({\phi_1}^{2}, \frac{1}{24}, \frac{-1}{2}\right), {\color{blue}{\phi_1}}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{1}{24}, \frac{-1}{2}\right), {\phi_1}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{1}{24}, \frac{-1}{2}\right), {\phi_1}^{2}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{1}{24}, \frac{-1}{2}\right), \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 99.0

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Applied rewrites 99.0%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), \phi_1 \cdot \phi_1, 1\right)} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\\ t_3 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, 1, t\_2\right) \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_1}\\ \mathbf{if}\;\phi_1 \leq -20:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, t\_2\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos lambda1) (- (sin lambda2))))
        (t_3
         (atan2
          (* (fma (sin lambda1) 1.0 t_2) (cos phi2))
          (- t_0 (* (* (sin phi1) (cos phi2)) t_1)))))
   (if (<= phi1 -20.0)
     t_3
     (if (<= phi1 19.0)
       (atan2
        (* (fma (sin lambda1) (cos lambda2) t_2) (cos phi2))
        (- t_0 (* (sin phi1) t_1)))
       t_3))))

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 = cos(lambda1) * -sin(lambda2);
	double t_3 = atan2((fma(sin(lambda1), 1.0, t_2) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * t_1)));
	double tmp;
	if (phi1 <= -20.0) {
		tmp = t_3;
	} else if (phi1 <= 19.0) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), t_2) * cos(phi2)), (t_0 - (sin(phi1) * t_1)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(lambda1) * Float64(-sin(lambda2)))
	t_3 = atan(Float64(fma(sin(lambda1), 1.0, t_2) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * t_1)))
	tmp = 0.0
	if (phi1 <= -20.0)
		tmp = t_3;
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), t_2) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * t_1)));
	else
		tmp = t_3;
	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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * 1.0 + t$95$2), $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]}, If[LessEqual[phi1, -20.0], t$95$3, If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -20 or 19 < phi1

   1. Initial program 76.7%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 79.6

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

   3. Applied rewrites 79.6%

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

   4. Taylor expanded in lambda2 around 0

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

   6. Applied rewrites 77.7%

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

   if -20 < phi1 < 19

   1. Initial program 81.4%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 98.7

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

   6. Applied rewrites 98.7%

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

Alternative 11: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -310:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - t\_1 \cdot t\_2}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-t\_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -310.0)
     (atan2 t_3 (- t_0 (* t_1 t_2)))
     (if (<= phi1 19.0)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
         (cos phi2))
        (- t_0 (* (sin phi1) t_2)))
       (atan2
        t_3
        (fma (sin phi2) (cos phi1) (* (- t_1) (cos (- lambda2 lambda1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -310.0) {
		tmp = atan2(t_3, (t_0 - (t_1 * t_2)));
	} else if (phi1 <= 19.0) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * t_2)));
	} else {
		tmp = atan2(t_3, fma(sin(phi2), cos(phi1), (-t_1 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -310.0)
		tmp = atan(t_3, Float64(t_0 - Float64(t_1 * t_2)));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * t_2)));
	else
		tmp = atan(t_3, fma(sin(phi2), cos(phi1), Float64(Float64(-t_1) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -310.0], N[ArcTan[t$95$3 / N[(t$95$0 - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$3 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[((-t$95$1) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -310

   1. Initial program 76.2%

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

   if -310 < phi1 < 19

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 19 < phi1

   1. Initial program 77.3%

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

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cos.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

   3. Applied rewrites 77.3%

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

Alternative 12: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -310:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-t\_1\right) \cdot t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2 (cos (- lambda2 lambda1)))
        (t_3 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -310.0)
     (atan2 t_3 (- t_0 (* t_1 (cos (- lambda1 lambda2)))))
     (if (<= phi1 19.0)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (- t_0 (* (sin phi1) t_2)))
       (atan2 t_3 (fma (sin phi2) (cos phi1) (* (- 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 = sin(phi1) * cos(phi2);
	double t_2 = cos((lambda2 - lambda1));
	double t_3 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -310.0) {
		tmp = atan2(t_3, (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	} else if (phi1 <= 19.0) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (sin(phi1) * t_2)));
	} else {
		tmp = atan2(t_3, fma(sin(phi2), cos(phi1), (-t_1 * t_2)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = cos(Float64(lambda2 - lambda1))
	t_3 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -310.0)
		tmp = atan(t_3, Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * t_2)));
	else
		tmp = atan(t_3, fma(sin(phi2), cos(phi1), Float64(Float64(-t_1) * 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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -310.0], N[ArcTan[t$95$3 / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$3 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[((-t$95$1) * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -310

   1. Initial program 76.2%

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

   if -310 < phi1 < 19

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-neg-rev N/A

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

      11. mul-1-neg N/A

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

      12. sin-sum N/A

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

      13. mul-1-neg N/A

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

      14. negate-sub N/A

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

      15. sin-diff N/A

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

      16. *-commutative N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-sin.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-diff-rev N/A

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

      2. cos-neg-rev N/A

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

      3. negate-sub2 N/A

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

      4. cos-diff-rev N/A

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

      5. negate-sub2 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 98.5

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

   10. Applied rewrites 98.5%

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

   if 19 < phi1

   1. Initial program 77.3%

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

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cos.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

   3. Applied rewrites 77.3%

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

Alternative 13: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot t\_1\right) + \sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-t\_0\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos phi2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -1.85e-5)
     (atan2
      t_2
      (- (* (cos phi1) (sin phi2)) (* t_0 (cos (- lambda1 lambda2)))))
     (if (<= phi1 19.0)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
         (cos phi2))
        (+ (- (* (* (cos phi2) phi1) t_1)) (sin phi2)))
       (atan2 t_2 (fma (sin phi2) (cos phi1) (* (- t_0) t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos(phi2);
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -1.85e-5) {
		tmp = atan2(t_2, ((cos(phi1) * sin(phi2)) - (t_0 * cos((lambda1 - lambda2)))));
	} else if (phi1 <= 19.0) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (-((cos(phi2) * phi1) * t_1) + sin(phi2)));
	} else {
		tmp = atan2(t_2, fma(sin(phi2), cos(phi1), (-t_0 * t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -1.85e-5)
		tmp = atan(t_2, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(Float64(-Float64(Float64(cos(phi2) * phi1) * t_1)) + sin(phi2)));
	else
		tmp = atan(t_2, fma(sin(phi2), cos(phi1), Float64(Float64(-t_0) * t_1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.85e-5], N[ArcTan[t$95$2 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[((-N[(N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision] * t$95$1), $MachinePrecision]) + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[((-t$95$0) * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.84999999999999991e-5

   1. Initial program 76.0%

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

   if -1.84999999999999991e-5 < phi1 < 19

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.4

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. cos-neg-rev N/A

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

      4. negate-sub2 N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. negate-sub N/A

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

      9. associate-*r* N/A

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

      10. lower-+.f64 N/A

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

   6. Applied rewrites 99.1%

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

   if 19 < phi1

   1. Initial program 77.3%

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

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cos.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

   3. Applied rewrites 77.3%

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

Alternative 14: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-t\_0\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -1.55e-5)
     (atan2
      t_1
      (- (* (cos phi1) (sin phi2)) (* t_0 (cos (- lambda1 lambda2)))))
     (if (<= phi1 19.0)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_1
        (fma (sin phi2) (cos phi1) (* (- t_0) (cos (- lambda2 lambda1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos(phi2);
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -1.55e-5) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (t_0 * cos((lambda1 - lambda2)))));
	} else if (phi1 <= 19.0) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, fma(sin(phi2), cos(phi1), (-t_0 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -1.55e-5)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, fma(sin(phi2), cos(phi1), Float64(Float64(-t_0) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.55e-5], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[((-t$95$0) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.55000000000000007e-5

   1. Initial program 76.0%

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

   if -1.55000000000000007e-5 < phi1 < 19

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.4

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

   3. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-neg-rev N/A

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

      11. mul-1-neg N/A

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

      12. sin-sum N/A

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

      13. mul-1-neg N/A

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

      14. negate-sub N/A

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

      15. sin-diff N/A

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

      16. *-commutative N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-sin.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. cos-neg-rev N/A

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

      3. negate-sub2 N/A

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

      4. lift-sin.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if 19 < phi1

   1. Initial program 77.3%

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

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cos.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

   3. Applied rewrites 77.3%

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

Alternative 15: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (* (sin (- lambda1 lambda2)) (cos phi2))
          (-
           (* (cos phi1) (sin phi2))
           (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
   (if (<= phi1 -1.55e-5)
     t_0
     (if (<= phi1 19.0)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -1.55e-5) {
		tmp = t_0;
	} else if (phi1 <= 19.0) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} 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 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    if (phi1 <= (-1.55d-5)) then
        tmp = t_0
    else if (phi1 <= 19.0d0) then
        tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    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 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -1.55e-5) {
		tmp = t_0;
	} else if (phi1 <= 19.0) {
		tmp = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = 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)))))
	tmp = 0
	if phi1 <= -1.55e-5:
		tmp = t_0
	elif phi1 <= 19.0:
		tmp = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi1 <= -1.55e-5)
		tmp = t_0;
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	tmp = 0.0;
	if (phi1 <= -1.55e-5)
		tmp = t_0;
	elseif (phi1 <= 19.0)
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = t_0;
	end
	tmp_2 = 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[(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[phi1, -1.55e-5], t$95$0, If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.55000000000000007e-5 or 19 < phi1

   1. Initial program 76.7%

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

   if -1.55000000000000007e-5 < phi1 < 19

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.4

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

   3. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-neg-rev N/A

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

      11. mul-1-neg N/A

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

      12. sin-sum N/A

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

      13. mul-1-neg N/A

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

      14. negate-sub N/A

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

      15. sin-diff N/A

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

      16. *-commutative N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-sin.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. cos-neg-rev N/A

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

      3. negate-sub2 N/A

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

      4. lift-sin.f64 95.6

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

   10. Applied rewrites 95.6%

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

Alternative 16: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.0075:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -1e+27)
     (atan2
      (*
       (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
       (cos phi2))
      (sin phi2))
     (if (<= lambda2 0.0075)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* (* (cos lambda1) (cos phi2)) (sin phi1))))
       (atan2
        (* (sin (- lambda2)) (cos phi2))
        (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else if (lambda2 <= 0.0075) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	} else {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if (lambda2 <= (-1d+27)) then
        tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    else if (lambda2 <= 0.0075d0) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))))
    else
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	} else if (lambda2 <= 0.0075) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - ((Math.cos(lambda1) * Math.cos(phi2)) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_0 - ((Math.cos(lambda2) * Math.sin(phi1)) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -1e+27:
		tmp = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	elif lambda2 <= 0.0075:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - ((math.cos(lambda1) * math.cos(phi2)) * math.sin(phi1))))
	else:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_0 - ((math.cos(lambda2) * math.sin(phi1)) * math.cos(phi2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -1e+27)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	elseif (lambda2 <= 0.0075)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(cos(lambda1) * cos(phi2)) * sin(phi1))));
	else
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -1e+27)
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	elseif (lambda2 <= 0.0075)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda1) * cos(phi2)) * sin(phi1))));
	else
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1e+27], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.0075], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1e27

   1. Initial program 60.6%

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

      1. lift--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. negate-sub 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. sin-sum-rev N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. sin-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 81.0

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

   3. Applied rewrites 81.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. negate-sub2 N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 60.2

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 60.2%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -1e27 < lambda2 < 0.0074999999999999997

   1. Initial program 97.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      6. lift-sin.f64 96.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 96.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   if 0.0074999999999999997 < lambda2

   1. Initial program 58.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 58.3

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 58.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      2. lower-neg.f64 58.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   7. Applied rewrites 58.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0 \cdot \cos \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (sin phi1))) (t_1 (* (cos phi1) (sin phi2))))
   (if (<= lambda1 -3.8e-7)
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_1 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     (if (<= lambda1 1.3e-185)
       (atan2 (* (sin (- lambda2)) (cos phi2)) (- t_1 (* t_0 (cos phi2))))
       (if (<= lambda1 2.7e-13)
         (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- t_1 t_0))
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (sin phi2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * sin(phi1);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -3.8e-7) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else if (lambda1 <= 1.3e-185) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))));
	} else if (lambda1 <= 2.7e-13) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0));
	} else {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(lambda2) * sin(phi1)
    t_1 = cos(phi1) * sin(phi2)
    if (lambda1 <= (-3.8d-7)) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else if (lambda1 <= 1.3d-185) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))))
    else if (lambda1 <= 2.7d-13) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0))
    else
        tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda2) * Math.sin(phi1);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -3.8e-7) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_1 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	} else if (lambda1 <= 1.3e-185) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_1 - (t_0 * Math.cos(phi2))));
	} else if (lambda1 <= 2.7e-13) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_1 - t_0));
	} else {
		tmp = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda2) * math.sin(phi1)
	t_1 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -3.8e-7:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_1 - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	elif lambda1 <= 1.3e-185:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_1 - (t_0 * math.cos(phi2))))
	elif lambda1 <= 2.7e-13:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_1 - t_0))
	else:
		tmp = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * sin(phi1))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -3.8e-7)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_1 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda1 <= 1.3e-185)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_1 - Float64(t_0 * cos(phi2))));
	elseif (lambda1 <= 2.7e-13)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_1 - t_0));
	else
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(lambda2) * sin(phi1);
	t_1 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -3.8e-7)
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	elseif (lambda1 <= 1.3e-185)
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))));
	elseif (lambda1 <= 2.7e-13)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0));
	else
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.8e-7], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 1.3e-185], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 2.7e-13], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -3.80000000000000015e-7

   1. Initial program 58.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 58.3

         \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 58.3%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if -3.80000000000000015e-7 < lambda1 < 1.29999999999999992e-185

   1. Initial program 99.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 99.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      2. lower-neg.f64 84.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   7. Applied rewrites 84.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   if 1.29999999999999992e-185 < lambda1 < 2.70000000000000011e-13

   1. Initial program 99.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 99.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      3. lift-*.f64 83.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

   7. Applied rewrites 83.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]

   if 2.70000000000000011e-13 < lambda1

   1. Initial program 59.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 79.9

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 79.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 60.5

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 60.5%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_1 \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0 \cdot \cos \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (sin phi1))) (t_1 (* (cos phi1) (sin phi2))))
   (if (<= lambda1 -3.8e-7)
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_1 (* (* (sin phi1) (cos phi2)) (cos lambda1))))
     (if (<= lambda1 1.3e-185)
       (atan2 (* (sin (- lambda2)) (cos phi2)) (- t_1 (* t_0 (cos phi2))))
       (if (<= lambda1 2.7e-13)
         (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- t_1 t_0))
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (sin phi2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * sin(phi1);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -3.8e-7) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	} else if (lambda1 <= 1.3e-185) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))));
	} else if (lambda1 <= 2.7e-13) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0));
	} else {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(lambda2) * sin(phi1)
    t_1 = cos(phi1) * sin(phi2)
    if (lambda1 <= (-3.8d-7)) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))))
    else if (lambda1 <= 1.3d-185) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))))
    else if (lambda1 <= 2.7d-13) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0))
    else
        tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda2) * Math.sin(phi1);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -3.8e-7) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_1 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else if (lambda1 <= 1.3e-185) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_1 - (t_0 * Math.cos(phi2))));
	} else if (lambda1 <= 2.7e-13) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_1 - t_0));
	} else {
		tmp = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda2) * math.sin(phi1)
	t_1 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -3.8e-7:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_1 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	elif lambda1 <= 1.3e-185:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_1 - (t_0 * math.cos(phi2))))
	elif lambda1 <= 2.7e-13:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_1 - t_0))
	else:
		tmp = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * sin(phi1))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -3.8e-7)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_1 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda1 <= 1.3e-185)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_1 - Float64(t_0 * cos(phi2))));
	elseif (lambda1 <= 2.7e-13)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_1 - t_0));
	else
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(lambda2) * sin(phi1);
	t_1 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -3.8e-7)
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda1 <= 1.3e-185)
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos(phi2))));
	elseif (lambda1 <= 2.7e-13)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - t_0));
	else
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.8e-7], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 1.3e-185], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 2.7e-13], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -3.80000000000000015e-7

   1. Initial program 58.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]

   if -3.80000000000000015e-7 < lambda1 < 1.29999999999999992e-185

   1. Initial program 99.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 99.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      2. lower-neg.f64 84.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   7. Applied rewrites 84.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   if 1.29999999999999992e-185 < lambda1 < 2.70000000000000011e-13

   1. Initial program 99.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 99.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      3. lift-*.f64 83.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

   7. Applied rewrites 83.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]

   if 2.70000000000000011e-13 < lambda1

   1. Initial program 59.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 79.9

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 79.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 60.5

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 60.5%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -8 \cdot 10^{-75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 1.75 \cdot 10^{-156}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+122}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (sin phi2)))
        (t_1 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -8e-75)
     t_0
     (if (<= lambda2 1.75e-156)
       (atan2
        (* (sin lambda1) (cos phi2))
        (- t_1 (* (* (sin phi1) (cos phi2)) (cos lambda1))))
       (if (<= lambda2 1.35e+122)
         (atan2
          (* (sin (- lambda1 lambda2)) (cos phi2))
          (- t_1 (* (sin phi1) (cos (- lambda1 lambda2)))))
         t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -8e-75) {
		tmp = t_0;
	} else if (lambda2 <= 1.75e-156) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	} else if (lambda2 <= 1.35e+122) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (sin(phi1) * cos((lambda1 - 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) :: t_1
    real(8) :: tmp
    t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    t_1 = cos(phi1) * sin(phi2)
    if (lambda2 <= (-8d-75)) then
        tmp = t_0
    else if (lambda2 <= 1.75d-156) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))))
    else if (lambda2 <= 1.35d+122) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (sin(phi1) * cos((lambda1 - 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.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -8e-75) {
		tmp = t_0;
	} else if (lambda2 <= 1.75e-156) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_1 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else if (lambda2 <= 1.35e+122) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_1 - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -8e-75:
		tmp = t_0
	elif lambda2 <= 1.75e-156:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_1 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	elif lambda2 <= 1.35e+122:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_1 - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -8e-75)
		tmp = t_0;
	elseif (lambda2 <= 1.75e-156)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_1 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda2 <= 1.35e+122)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_1 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	t_1 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -8e-75)
		tmp = t_0;
	elseif (lambda2 <= 1.75e-156)
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_1 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda2 <= 1.35e+122)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	tmp_2 = 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[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -8e-75], t$95$0, If[LessEqual[lambda2, 1.75e-156], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 1.35e+122], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -7.9999999999999997e-75 or 1.3499999999999999e122 < lambda2

   1. Initial program 64.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 82.5

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 82.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 59.4

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 59.4%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -7.9999999999999997e-75 < lambda2 < 1.75e-156

   1. Initial program 99.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 86.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]

   if 1.75e-156 < lambda2 < 1.3499999999999999e122

   1. Initial program 80.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lift-sin.f64 67.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 67.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 71.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, t\_0\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t\_0 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (sin lambda1))))
   (if (<= phi2 -0.003)
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos phi2))))
     (if (<= phi2 2.8e-16)
       (atan2
        (fma -1.0 (* (cos lambda1) (sin lambda2)) t_0)
        (* (- (cos (- lambda2 lambda1))) (sin phi1)))
       (atan2
        (* (- t_0 (* (sin lambda2) (cos lambda1))) (cos phi2))
        (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * sin(lambda1);
	double tmp;
	if (phi2 <= -0.003) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos(phi2))));
	} else if (phi2 <= 2.8e-16) {
		tmp = atan2(fma(-1.0, (cos(lambda1) * sin(lambda2)), t_0), (-cos((lambda2 - lambda1)) * sin(phi1)));
	} else {
		tmp = atan2(((t_0 - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * sin(lambda1))
	tmp = 0.0
	if (phi2 <= -0.003)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(phi2))));
	elseif (phi2 <= 2.8e-16)
		tmp = atan(fma(-1.0, Float64(cos(lambda1) * sin(lambda2)), t_0), Float64(Float64(-cos(Float64(lambda2 - lambda1))) * sin(phi1)));
	else
		tmp = atan(Float64(Float64(t_0 - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.003], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2.8e-16], N[ArcTan[N[(-1.0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[((-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]) * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$0 - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0030000000000000001

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 66.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 66.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\phi_2}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 57.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \phi_2} \]

   7. Applied rewrites 57.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\phi_2}} \]

   if -0.0030000000000000001 < phi2 < 2.8000000000000001e-16

   1. Initial program 81.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 89.3

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.3%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-1 \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right) \cdot \sin \phi_1} \]

      3. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      4. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \sin \color{blue}{\phi_1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \color{blue}{\sin \phi_1}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \color{blue}{\phi_1}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      8. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      9. lift-sin.f64 86.2

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

   6. Applied rewrites 86.2%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1}} \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_2 \cdot \sin \lambda_1}}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]
   8. Step-by-step derivation

      1. cos-+PI/2-rev N/A

         \[\leadsto \tan^{-1}_* \frac{-1 \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_2 \cdot \sin \lambda_1}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \color{blue}{\cos \lambda_1 \cdot \sin \lambda_2}, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \color{blue}{\sin \lambda_2}, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \color{blue}{\lambda_2}, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      5. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      6. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      7. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

      8. lift-*.f64 86.2

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

   9. Applied rewrites 86.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-1, \cos \lambda_1 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \sin \lambda_1\right)}}{\left(-\cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1} \]

   if 2.8000000000000001e-16 < phi2

   1. Initial program 77.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 89.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.6

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 62.3

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 62.3%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot t\_0}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -1.55e-5)
     (atan2 t_1 (- (* (cos phi1) (sin phi2)) (* (sin phi1) t_0)))
     (if (<= phi1 19.0)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2 t_1 (- (sin phi2) (* (* (sin phi1) (cos phi2)) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -1.55e-5) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)));
	} else if (phi1 <= 19.0) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, (sin(phi2) - ((sin(phi1) * cos(phi2)) * 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) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin((lambda1 - lambda2)) * cos(phi2)
    if (phi1 <= (-1.55d-5)) then
        tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)))
    else if (phi1 <= 19.0d0) then
        tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2(t_1, (sin(phi2) - ((sin(phi1) * cos(phi2)) * 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.cos((lambda1 - lambda2));
	double t_1 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double tmp;
	if (phi1 <= -1.55e-5) {
		tmp = Math.atan2(t_1, ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * t_0)));
	} else if (phi1 <= 19.0) {
		tmp = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - ((Math.sin(phi1) * Math.cos(phi2)) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	tmp = 0
	if phi1 <= -1.55e-5:
		tmp = math.atan2(t_1, ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * t_0)))
	elif phi1 <= 19.0:
		tmp = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2(t_1, (math.sin(phi2) - ((math.sin(phi1) * math.cos(phi2)) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -1.55e-5)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * t_0)));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, Float64(sin(phi2) - Float64(Float64(sin(phi1) * cos(phi2)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	tmp = 0.0;
	if (phi1 <= -1.55e-5)
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)));
	elseif (phi1 <= 19.0)
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2(t_1, (sin(phi2) - ((sin(phi1) * cos(phi2)) * t_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[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.55e-5], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.55000000000000007e-5

   1. Initial program 76.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lift-sin.f64 50.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if -1.55000000000000007e-5 < phi1 < 19

   1. Initial program 81.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 99.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 95.6

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 95.6%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if 19 < phi1

   1. Initial program 77.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. lift-sin.f64 49.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 49.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+122}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (sin phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -1e+27)
     t_0
     (if (<= lambda2 6.8e-7)
       (atan2 t_1 (- t_2 (* (cos lambda1) (sin phi1))))
       (if (<= lambda2 1.35e+122)
         (atan2 t_1 (- t_2 (* (cos lambda2) (sin phi1))))
         t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_2 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = t_0;
	} else if (lambda2 <= 6.8e-7) {
		tmp = atan2(t_1, (t_2 - (cos(lambda1) * sin(phi1))));
	} else if (lambda2 <= 1.35e+122) {
		tmp = atan2(t_1, (t_2 - (cos(lambda2) * sin(phi1))));
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    t_1 = sin((lambda1 - lambda2)) * cos(phi2)
    t_2 = cos(phi1) * sin(phi2)
    if (lambda2 <= (-1d+27)) then
        tmp = t_0
    else if (lambda2 <= 6.8d-7) then
        tmp = atan2(t_1, (t_2 - (cos(lambda1) * sin(phi1))))
    else if (lambda2 <= 1.35d+122) then
        tmp = atan2(t_1, (t_2 - (cos(lambda2) * sin(phi1))))
    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.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	double t_1 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double t_2 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = t_0;
	} else if (lambda2 <= 6.8e-7) {
		tmp = Math.atan2(t_1, (t_2 - (Math.cos(lambda1) * Math.sin(phi1))));
	} else if (lambda2 <= 1.35e+122) {
		tmp = Math.atan2(t_1, (t_2 - (Math.cos(lambda2) * Math.sin(phi1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	t_1 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	t_2 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -1e+27:
		tmp = t_0
	elif lambda2 <= 6.8e-7:
		tmp = math.atan2(t_1, (t_2 - (math.cos(lambda1) * math.sin(phi1))))
	elif lambda2 <= 1.35e+122:
		tmp = math.atan2(t_1, (t_2 - (math.cos(lambda2) * math.sin(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(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -1e+27)
		tmp = t_0;
	elseif (lambda2 <= 6.8e-7)
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda1) * sin(phi1))));
	elseif (lambda2 <= 1.35e+122)
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda2) * sin(phi1))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	t_2 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -1e+27)
		tmp = t_0;
	elseif (lambda2 <= 6.8e-7)
		tmp = atan2(t_1, (t_2 - (cos(lambda1) * sin(phi1))));
	elseif (lambda2 <= 1.35e+122)
		tmp = atan2(t_1, (t_2 - (cos(lambda2) * sin(phi1))));
	else
		tmp = t_0;
	end
	tmp_2 = 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[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1e+27], t$95$0, If[LessEqual[lambda2, 6.8e-7], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 1.35e+122], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1e27 or 1.3499999999999999e122 < lambda2

   1. Initial program 59.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 80.6

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 80.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 60.2

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 60.2%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -1e27 < lambda2 < 6.79999999999999948e-7

   1. Initial program 97.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \phi_1}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\phi_1}} \]

      4. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

      6. lift-sin.f64 96.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 96.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \color{blue}{\phi_1}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 80.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1} \]

   7. Applied rewrites 80.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \color{blue}{\phi_1}} \]

   if 6.79999999999999948e-7 < lambda2 < 1.3499999999999999e122

   1. Initial program 58.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 58.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 58.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

      3. lift-*.f64 50.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1} \]

   7. Applied rewrites 50.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+122}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
           (cos phi2))
          (sin phi2))))
   (if (<= lambda2 -1e+27)
     t_0
     (if (<= lambda2 1.35e+122)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- (sin phi2) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = t_0;
	} else if (lambda2 <= 1.35e+122) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - 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((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
    if (lambda2 <= (-1d+27)) then
        tmp = t_0
    else if (lambda2 <= 1.35d+122) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - 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.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), Math.sin(phi2));
	double tmp;
	if (lambda2 <= -1e+27) {
		tmp = t_0;
	} else if (lambda2 <= 1.35e+122) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (Math.sin(phi2) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), math.sin(phi2))
	tmp = 0
	if lambda2 <= -1e+27:
		tmp = t_0
	elif lambda2 <= 1.35e+122:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (math.sin(phi2) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (lambda2 <= -1e+27)
		tmp = t_0;
	elseif (lambda2 <= 1.35e+122)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(sin(phi2) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	tmp = 0.0;
	if (lambda2 <= -1e+27)
		tmp = t_0;
	elseif (lambda2 <= 1.35e+122)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	tmp_2 = 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[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -1e+27], t$95$0, If[LessEqual[lambda2, 1.35e+122], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1e27 or 1.3499999999999999e122 < lambda2

   1. Initial program 59.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 80.6

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 80.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 60.2

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 60.2%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -1e27 < lambda2 < 1.3499999999999999e122

   1. Initial program 90.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. lift-sin.f64 76.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 76.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq -0.0029:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 0.07:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_2}{t\_0 - \left(\sin \phi_1 \cdot t\_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 -0.0029)
     (atan2 (* t_1 (cos phi2)) (- t_0 (* (sin phi1) (cos phi2))))
     (if (<= phi2 0.07)
       (atan2
        (* t_1 t_2)
        (- t_0 (* (* (sin phi1) t_2) (cos (- lambda1 lambda2)))))
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= -0.0029) {
		tmp = atan2((t_1 * cos(phi2)), (t_0 - (sin(phi1) * cos(phi2))));
	} else if (phi2 <= 0.07) {
		tmp = atan2((t_1 * t_2), (t_0 - ((sin(phi1) * t_2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= -0.0029)
		tmp = atan(Float64(t_1 * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(phi2))));
	elseif (phi2 <= 0.07)
		tmp = atan(Float64(t_1 * t_2), Float64(t_0 - Float64(Float64(sin(phi1) * t_2) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, -0.0029], N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.07], N[ArcTan[N[(t$95$1 * t$95$2), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0029

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \color{blue}{\cos \phi_2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \phi_1\right) \cdot \cos \color{blue}{\phi_2}} \]

      4. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      6. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

      7. lift-cos.f64 66.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2} \]

   4. Applied rewrites 66.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}} \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\phi_2}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 57.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \phi_2} \]

   7. Applied rewrites 57.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\phi_2}} \]

   if -0.0029 < phi2 < 0.070000000000000007

   1. Initial program 81.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 81.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 81.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 81.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 81.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 0.070000000000000007 < phi2

   1. Initial program 77.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 89.7

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 89.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.6

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 62.3

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 62.3%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0)) (t_1 (sin (- lambda1 lambda2))))
   (if (<= phi1 -5e-5)
     (atan2 (* t_1 (cos phi2)) (- (* (cos (- lambda2 lambda1)) (sin phi1))))
     (if (<= phi1 19.0)
       (atan2
        (*
         (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        (* t_1 t_0)
        (-
         (* (cos phi1) (sin phi2))
         (* (* (sin phi1) t_0) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5e-5) {
		tmp = atan2((t_1 * cos(phi2)), -(cos((lambda2 - lambda1)) * sin(phi1)));
	} else if (phi1 <= 19.0) {
		tmp = atan2((((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((t_1 * t_0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * t_0) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -5e-5)
		tmp = atan(Float64(t_1 * cos(phi2)), Float64(-Float64(cos(Float64(lambda2 - lambda1)) * sin(phi1))));
	elseif (phi1 <= 19.0)
		tmp = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(t_1 * t_0), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * t_0) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5e-5], N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / (-N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])], $MachinePrecision], If[LessEqual[phi1, 19.0], N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.00000000000000024e-5

   1. Initial program 76.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \sin \phi_1} \]

      5. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      8. lift-sin.f64 47.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 47.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]

   if -5.00000000000000024e-5 < phi1 < 19

   1. Initial program 81.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. negate-sub 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. sin-sum-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. sin-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-sin.f64 99.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\color{blue}{\sin \lambda_2}\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)}} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}} \]

      5. cos-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)} \]

      11. lift-sin.f64 99.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1} \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_2\right)} + \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \color{blue}{\cos \lambda_1} \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      10. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \left(-1 \cdot \lambda_2\right) + \cos \lambda_1 \cdot \sin \color{blue}{\left(-1 \cdot \lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      12. sin-sum N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      14. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      15. sin-diff N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\cos \lambda_2} \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \color{blue}{\sin \lambda_1} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      21. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. cos-diff-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lift-sin.f64 95.6

         \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 95.6%

       \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if 19 < phi1

   1. Initial program 77.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 43.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 43.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 44.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 44.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0029:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 0.145:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (atan2 (* t_1 (cos phi2)) (sin phi2))))
   (if (<= phi2 -0.0029)
     t_2
     (if (<= phi2 0.145)
       (atan2
        (* t_1 t_0)
        (-
         (* (cos phi1) (sin phi2))
         (* (* (sin phi1) t_0) (cos (- lambda1 lambda2)))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = atan2((t_1 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -0.0029) {
		tmp = t_2;
	} else if (phi2 <= 0.145) {
		tmp = atan2((t_1 * t_0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * t_0) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = atan(Float64(t_1 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.0029)
		tmp = t_2;
	elseif (phi2 <= 0.145)
		tmp = atan(Float64(t_1 * t_0), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * t_0) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.0029], t$95$2, If[LessEqual[phi2, 0.145], N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0029 or 0.14499999999999999 < phi2

   1. Initial program 76.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.0029 < phi2 < 0.14499999999999999

   1. Initial program 81.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 81.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 81.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. lower-*.f64 81.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 81.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 65.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0053:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.023:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* t_0 (cos phi2)) (sin phi2))))
   (if (<= phi2 -0.0053)
     t_1
     (if (<= phi2 0.023)
       (atan2
        (* t_0 1.0)
        (- (* phi2 (cos phi1)) (* (cos (- lambda2 lambda1)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -0.0053) {
		tmp = t_1;
	} else if (phi2 <= 0.023) {
		tmp = atan2((t_0 * 1.0), ((phi2 * cos(phi1)) - (cos((lambda2 - lambda1)) * 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)), sin(phi2))
    if (phi2 <= (-0.0053d0)) then
        tmp = t_1
    else if (phi2 <= 0.023d0) then
        tmp = atan2((t_0 * 1.0d0), ((phi2 * cos(phi1)) - (cos((lambda2 - lambda1)) * 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)), Math.sin(phi2));
	double tmp;
	if (phi2 <= -0.0053) {
		tmp = t_1;
	} else if (phi2 <= 0.023) {
		tmp = Math.atan2((t_0 * 1.0), ((phi2 * Math.cos(phi1)) - (Math.cos((lambda2 - lambda1)) * 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)), math.sin(phi2))
	tmp = 0
	if phi2 <= -0.0053:
		tmp = t_1
	elif phi2 <= 0.023:
		tmp = math.atan2((t_0 * 1.0), ((phi2 * math.cos(phi1)) - (math.cos((lambda2 - lambda1)) * 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)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.0053)
		tmp = t_1;
	elseif (phi2 <= 0.023)
		tmp = atan(Float64(t_0 * 1.0), Float64(Float64(phi2 * cos(phi1)) - Float64(cos(Float64(lambda2 - lambda1)) * 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)), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -0.0053)
		tmp = t_1;
	elseif (phi2 <= 0.023)
		tmp = atan2((t_0 * 1.0), ((phi2 * cos(phi1)) - (cos((lambda2 - lambda1)) * 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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.0053], t$95$1, If[LessEqual[phi2, 0.023], N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.00530000000000000002 or 0.023 < phi2

   1. Initial program 76.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.00530000000000000002 < phi2 < 0.023

   1. Initial program 81.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

       3. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

       4. negate-sub2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \sin \phi_1} \]

       5. cos-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \color{blue}{\phi_1}} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

       7. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \color{blue}{\phi_1}} \]

       8. lift--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

       9. lift-sin.f64 81.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

   13. Applied rewrites 81.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 64.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_1}{-t\_0 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_1 \leq -0.047:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 19:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-t\_0, \phi_1, \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (atan2 t_1 (- (* t_0 (sin phi1))))))
   (if (<= phi1 -0.047)
     t_2
     (if (<= phi1 19.0) (atan2 t_1 (fma (- t_0) phi1 (sin phi2))) t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_2 = atan2(t_1, -(t_0 * sin(phi1)));
	double tmp;
	if (phi1 <= -0.047) {
		tmp = t_2;
	} else if (phi1 <= 19.0) {
		tmp = atan2(t_1, fma(-t_0, phi1, sin(phi2)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_2 = atan(t_1, Float64(-Float64(t_0 * sin(phi1))))
	tmp = 0.0
	if (phi1 <= -0.047)
		tmp = t_2;
	elseif (phi1 <= 19.0)
		tmp = atan(t_1, fma(Float64(-t_0), phi1, sin(phi2)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / (-N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[phi1, -0.047], t$95$2, If[LessEqual[phi1, 19.0], N[ArcTan[t$95$1 / N[((-t$95$0) * phi1 + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.047 or 19 < phi1

   1. Initial program 76.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. negate-sub2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \sin \phi_1} \]

      5. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

      8. lift-sin.f64 47.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1} \]

   4. Applied rewrites 47.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]

   if -0.047 < phi1 < 19

   1. Initial program 81.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + \phi_1 \cdot \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \phi_2\right) - \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \phi_2\right) - \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\sin \phi_2}} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \phi_2\right) - \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \phi_1 + \sin \color{blue}{\phi_2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \phi_2\right) - \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \color{blue}{\phi_1}, \sin \phi_2\right)} \]

   4. Applied rewrites 81.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\left(\sin \phi_2 \cdot \phi_1\right) \cdot -0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \phi_1, \sin \phi_2\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \phi_1, \sin \phi_2\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{neg}\left(\cos \left(\lambda_2 - \lambda_1\right)\right), \phi_1, \sin \phi_2\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \left(\lambda_2 - \lambda_1\right), \phi_1, \sin \phi_2\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \left(\lambda_2 - \lambda_1\right), \phi_1, \sin \phi_2\right)} \]

      4. lift--.f64 80.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \left(\lambda_2 - \lambda_1\right), \phi_1, \sin \phi_2\right)} \]

   7. Applied rewrites 80.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \left(\lambda_2 - \lambda_1\right), \phi_1, \sin \phi_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* t_0 (cos phi2)) (sin phi2))))
   (if (<= phi2 -0.003)
     t_1
     (if (<= phi2 6e-15)
       (atan2 (* t_0 1.0) (* -1.0 (* (cos (- lambda2 lambda1)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = t_1;
	} else if (phi2 <= 6e-15) {
		tmp = atan2((t_0 * 1.0), (-1.0 * (cos((lambda2 - lambda1)) * 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)), sin(phi2))
    if (phi2 <= (-0.003d0)) then
        tmp = t_1
    else if (phi2 <= 6d-15) then
        tmp = atan2((t_0 * 1.0d0), ((-1.0d0) * (cos((lambda2 - lambda1)) * 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)), Math.sin(phi2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = t_1;
	} else if (phi2 <= 6e-15) {
		tmp = Math.atan2((t_0 * 1.0), (-1.0 * (Math.cos((lambda2 - lambda1)) * 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)), math.sin(phi2))
	tmp = 0
	if phi2 <= -0.003:
		tmp = t_1
	elif phi2 <= 6e-15:
		tmp = math.atan2((t_0 * 1.0), (-1.0 * (math.cos((lambda2 - lambda1)) * 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)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.003)
		tmp = t_1;
	elseif (phi2 <= 6e-15)
		tmp = atan(Float64(t_0 * 1.0), Float64(-1.0 * Float64(cos(Float64(lambda2 - lambda1)) * 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)), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -0.003)
		tmp = t_1;
	elseif (phi2 <= 6e-15)
		tmp = atan2((t_0 * 1.0), (-1.0 * (cos((lambda2 - lambda1)) * 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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.003], t$95$1, If[LessEqual[phi2, 6e-15], N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / N[(-1.0 * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0030000000000000001 or 6e-15 < phi2

   1. Initial program 76.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.0030000000000000001 < phi2 < 6e-15

   1. Initial program 81.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

       2. negate-sub2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \sin \phi_1\right)} \]

       3. cos-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}\right)} \]

       5. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \color{blue}{\phi_1}\right)} \]

       6. lift--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1\right)} \]

       7. lift-sin.f64 78.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1\right)} \]

   13. Applied rewrites 78.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 49.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Step-by-step derivation

   1. lift-sin.f64 49.2

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 49.2%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
5. Add Preprocessing

Alternative 31: 44.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_1 \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (atan2 (* (sin lambda1) (cos phi2)) (sin phi2))))
   (if (<= lambda1 -1.4)
     t_0
     (if (<= lambda1 2.5e-41)
       (atan2 (* (sin (- lambda2)) (cos phi2)) (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	double tmp;
	if (lambda1 <= -1.4) {
		tmp = t_0;
	} else if (lambda1 <= 2.5e-41) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2));
	} 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)), sin(phi2))
    if (lambda1 <= (-1.4d0)) then
        tmp = t_0
    else if (lambda1 <= 2.5d-41) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2))
    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.sin(phi2));
	double tmp;
	if (lambda1 <= -1.4) {
		tmp = t_0;
	} else if (lambda1 <= 2.5e-41) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2((math.sin(lambda1) * math.cos(phi2)), math.sin(phi2))
	tmp = 0
	if lambda1 <= -1.4:
		tmp = t_0
	elif lambda1 <= 2.5e-41:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(sin(lambda1) * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (lambda1 <= -1.4)
		tmp = t_0;
	elseif (lambda1 <= 2.5e-41)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	tmp = 0.0;
	if (lambda1 <= -1.4)
		tmp = t_0;
	elseif (lambda1 <= 2.5e-41)
		tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2));
	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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -1.4], t$95$0, If[LessEqual[lambda1, 2.5e-41], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.3999999999999999 or 2.4999999999999998e-41 < lambda1

   1. Initial program 60.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 40.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 40.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]

   if -1.3999999999999999 < lambda1 < 2.4999999999999998e-41

   1. Initial program 99.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 58.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 58.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. lower-neg.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 38.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.75 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.75e+16)
   (atan2
    (* (sin (- lambda1 lambda2)) (cos phi2))
    (*
     (fma
      (fma (* phi2 phi2) 0.008333333333333333 -0.16666666666666666)
      (* phi2 phi2)
      1.0)
     phi2))
   (atan2 (* (sin lambda1) (cos phi2)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.75e+16) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (fma(fma((phi2 * phi2), 0.008333333333333333, -0.16666666666666666), (phi2 * phi2), 1.0) * phi2));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.75e+16)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(fma(fma(Float64(phi2 * phi2), 0.008333333333333333, -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.75e+16], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.75e16

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      6. negate-sub N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      10. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

      13. lower-*.f64 41.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 41.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]

   if 1.75e16 < phi2

   1. Initial program 77.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 50.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 50.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 37.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -3.6:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), \phi_2 \cdot \phi_2, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi2 -3.6)
     (atan2 t_0 phi2)
     (atan2
      t_0
      (*
       (fma
        (fma
         (fma (* phi2 phi2) -0.0001984126984126984 0.008333333333333333)
         (* phi2 phi2)
         -0.16666666666666666)
        (* phi2 phi2)
        1.0)
       phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi2 <= -3.6) {
		tmp = atan2(t_0, phi2);
	} else {
		tmp = atan2(t_0, (fma(fma(fma((phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), (phi2 * phi2), -0.16666666666666666), (phi2 * phi2), 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -3.6)
		tmp = atan(t_0, phi2);
	else
		tmp = atan(t_0, Float64(fma(fma(fma(Float64(phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), Float64(phi2 * phi2), -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2));
	end
	return 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[phi2, -3.6], N[ArcTan[t$95$0 / phi2], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.60000000000000009

   1. Initial program 76.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 47.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 26.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]

   if -3.60000000000000009 < phi2

   1. Initial program 80.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \phi_2} \]

   7. Applied rewrites 41.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), \phi_2 \cdot \phi_2, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 37.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq 4400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi2 4400000000000.0)
     (atan2
      t_0
      (*
       (fma
        (fma (* phi2 phi2) 0.008333333333333333 -0.16666666666666666)
        (* phi2 phi2)
        1.0)
       phi2))
     (atan2 t_0 (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi2 <= 4400000000000.0) {
		tmp = atan2(t_0, (fma(fma((phi2 * phi2), 0.008333333333333333, -0.16666666666666666), (phi2 * phi2), 1.0) * phi2));
	} else {
		tmp = atan2(t_0, (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi2 <= 4400000000000.0)
		tmp = atan(t_0, Float64(fma(fma(Float64(phi2 * phi2), 0.008333333333333333, -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2));
	else
		tmp = atan(t_0, Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return 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[phi2, 4400000000000.0], N[ArcTan[t$95$0 / N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.4e12

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      6. negate-sub N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      10. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

      13. lower-*.f64 41.2

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 41.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]

   if 4.4e12 < phi2

   1. Initial program 76.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 26.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 26.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 35: 37.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq 4400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi2 4400000000000.0)
     (atan2 t_0 phi2)
     (atan2 t_0 (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi2 <= 4400000000000.0) {
		tmp = atan2(t_0, phi2);
	} else {
		tmp = atan2(t_0, (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi2 <= 4400000000000.0)
		tmp = atan(t_0, phi2);
	else
		tmp = atan(t_0, Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return 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[phi2, 4400000000000.0], N[ArcTan[t$95$0 / phi2], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.4e12

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]

   if 4.4e12 < phi2

   1. Initial program 76.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 26.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 26.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 36: 36.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.6 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.6e+19)
   (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) phi2)
   (atan2
    (* (+ lambda1 (* lambda2 (fma 0.5 (* lambda1 lambda1) -1.0))) (cos phi2))
    (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.6e+19) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), phi2);
	} else {
		tmp = atan2(((lambda1 + (lambda2 * fma(0.5, (lambda1 * lambda1), -1.0))) * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.6e+19)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), phi2);
	else
		tmp = atan(Float64(Float64(lambda1 + Float64(lambda2 * fma(0.5, Float64(lambda1 * lambda1), -1.0))) * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.6e+19], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[(lambda1 + N[(lambda2 * N[(0.5 * N[(lambda1 * lambda1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.6e19

   1. Initial program 79.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]

   if 1.6e19 < phi2

   1. Initial program 77.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.5

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       3. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       5. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, {\lambda_1}^{\color{blue}{2}}, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       6. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       7. lower-*.f64 21.9

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   13. Applied rewrites 21.9%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 37: 34.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6 \cdot 10^{-119}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\phi_2}\\ \mathbf{elif}\;\phi_2 \leq 4400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -6e-119)
   (atan2 (* (cos phi2) (- (sin lambda2))) phi2)
   (if (<= phi2 4400000000000.0)
     (atan2 (sin (- lambda1 lambda2)) phi2)
     (atan2
      (* (+ lambda1 (* lambda2 (fma 0.5 (* lambda1 lambda1) -1.0))) (cos phi2))
      (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -6e-119) {
		tmp = atan2((cos(phi2) * -sin(lambda2)), phi2);
	} else if (phi2 <= 4400000000000.0) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2(((lambda1 + (lambda2 * fma(0.5, (lambda1 * lambda1), -1.0))) * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -6e-119)
		tmp = atan(Float64(cos(phi2) * Float64(-sin(lambda2))), phi2);
	elseif (phi2 <= 4400000000000.0)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(Float64(Float64(lambda1 + Float64(lambda2 * fma(0.5, Float64(lambda1 * lambda1), -1.0))) * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -6e-119], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision] / phi2], $MachinePrecision], If[LessEqual[phi2, 4400000000000.0], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[(lambda1 + N[(lambda2 * N[(0.5 * N[(lambda1 * lambda1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.0000000000000004e-119

   1. Initial program 77.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\phi_2} \]
   12. Step-by-step derivation

       1. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}{\phi_2} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}}{\phi_2} \]

       3. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \lambda_2}\right)\right)}{\phi_2} \]

       4. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}{\phi_2} \]

       5. lift-neg.f64 25.8

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 25.8%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}}{\phi_2} \]

   if -6.0000000000000004e-119 < phi2 < 4.4e12

   1. Initial program 81.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 48.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

       2. lift--.f64 47.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]

   if 4.4e12 < phi2

   1. Initial program 76.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       3. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       5. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, {\lambda_1}^{\color{blue}{2}}, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       6. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       7. lower-*.f64 21.8

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   13. Applied rewrites 21.8%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 38: 33.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{+28}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\phi_2}\\ \mathbf{elif}\;\phi_2 \leq 4400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.35e+28)
   (atan2 (* (cos phi2) (sin lambda1)) phi2)
   (if (<= phi2 4400000000000.0)
     (atan2 (sin (- lambda1 lambda2)) phi2)
     (atan2
      (* (+ lambda1 (* lambda2 (fma 0.5 (* lambda1 lambda1) -1.0))) (cos phi2))
      (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.35e+28) {
		tmp = atan2((cos(phi2) * sin(lambda1)), phi2);
	} else if (phi2 <= 4400000000000.0) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2(((lambda1 + (lambda2 * fma(0.5, (lambda1 * lambda1), -1.0))) * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.35e+28)
		tmp = atan(Float64(cos(phi2) * sin(lambda1)), phi2);
	elseif (phi2 <= 4400000000000.0)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(Float64(Float64(lambda1 + Float64(lambda2 * fma(0.5, Float64(lambda1 * lambda1), -1.0))) * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.35e+28], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / phi2], $MachinePrecision], If[LessEqual[phi2, 4400000000000.0], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[(lambda1 + N[(lambda2 * N[(0.5 * N[(lambda1 * lambda1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.3500000000000001e28

   1. Initial program 76.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 47.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \sin \lambda_1}}{\phi_2} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\sin \lambda_1}}{\phi_2} \]

       2. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\lambda_1}}{\phi_2} \]

       3. lift-sin.f64 20.3

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\phi_2} \]

   13. Applied rewrites 20.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \sin \lambda_1}}{\phi_2} \]

   if -1.3500000000000001e28 < phi2 < 4.4e12

   1. Initial program 81.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

       2. lift--.f64 46.9

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 46.9%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]

   if 4.4e12 < phi2

   1. Initial program 76.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       3. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       5. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, {\lambda_1}^{\color{blue}{2}}, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       6. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       7. lower-*.f64 21.8

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   13. Applied rewrites 21.8%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 39: 33.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4400000000000.0)
   (atan2 (sin (- lambda1 lambda2)) phi2)
   (atan2
    (* (+ lambda1 (* lambda2 (fma 0.5 (* lambda1 lambda1) -1.0))) (cos phi2))
    (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4400000000000.0) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2(((lambda1 + (lambda2 * fma(0.5, (lambda1 * lambda1), -1.0))) * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4400000000000.0)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(Float64(Float64(lambda1 + Float64(lambda2 * fma(0.5, Float64(lambda1 * lambda1), -1.0))) * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4400000000000.0], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[(lambda1 + N[(lambda2 * N[(0.5 * N[(lambda1 * lambda1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.4e12

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

       2. lift--.f64 37.3

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 37.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]

   if 4.4e12 < phi2

   1. Initial program 76.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       3. negate-sub N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2} + -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       5. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, {\lambda_1}^{\color{blue}{2}}, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       6. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(\frac{1}{2}, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

       7. lower-*.f64 21.8

          \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   13. Applied rewrites 21.8%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 + \color{blue}{\lambda_2 \cdot \mathsf{fma}\left(0.5, \lambda_1 \cdot \lambda_1, -1\right)}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 40: 32.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.6:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.6)
   (atan2
    (* (sin (- lambda1 lambda2)) 1.0)
    (*
     (fma
      (fma (* phi2 phi2) 0.008333333333333333 -0.16666666666666666)
      (* phi2 phi2)
      1.0)
     phi2))
   (atan2
    (* lambda1 (cos phi2))
    (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.6) {
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), (fma(fma((phi2 * phi2), 0.008333333333333333, -0.16666666666666666), (phi2 * phi2), 1.0) * phi2));
	} else {
		tmp = atan2((lambda1 * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.6)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * 1.0), Float64(fma(fma(Float64(phi2 * phi2), 0.008333333333333333, -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2));
	else
		tmp = atan(Float64(lambda1 * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.6], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.6000000000000001

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\left(\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      6. negate-sub N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      10. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

      13. lower-*.f64 37.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 37.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]

   if 1.6000000000000001 < phi2

   1. Initial program 77.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 50.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 50.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

   13. Applied rewrites 18.3%

       \[\leadsto \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 41: 32.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.6:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.6)
   (atan2 (sin (- lambda1 lambda2)) phi2)
   (atan2
    (* lambda1 (cos phi2))
    (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.6) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2((lambda1 * cos(phi2)), (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.6)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(Float64(lambda1 * cos(phi2)), Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.6], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.6000000000000001

   1. Initial program 79.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 49.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 49.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

       2. lift--.f64 37.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 37.7%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]

   if 1.6000000000000001 < phi2

   1. Initial program 77.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   3. Step-by-step derivation

      1. lift-sin.f64 50.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 50.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_1 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot \lambda_1 + \sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \color{blue}{\lambda_1}, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. cos-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_2 + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \lambda_1\right) \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \lambda_2, \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. sin-neg-rev N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right), \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lift-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      12. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      13. sin-neg-rev N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      14. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, \mathsf{neg}\left(\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      15. lift-neg.f64 37.4

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 37.4%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \phi_2} \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]

      7. lower-*.f64 23.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   10. Applied rewrites 23.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \lambda_1, -\sin \lambda_2, \cos \lambda_2\right), \lambda_1, -\sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2} \]
   12. Step-by-step derivation

   13. Applied rewrites 18.3%

       \[\leadsto \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 42: 29.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) phi2))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), phi2);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin((lambda1 - lambda2)), phi2)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), phi2);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), phi2)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), phi2)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), phi2);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Step-by-step derivation

   1. lift-sin.f64 49.2

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 49.2%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]
6. Step-by-step derivation

7. Applied rewrites 32.2%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

8. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]
9. Step-by-step derivation

10. Applied rewrites 29.4%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

11. Taylor expanded in phi2 around 0

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

    2. lift--.f64 29.4

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

13. Applied rewrites 29.4%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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))))))