Bearing on a great circle

Percentage Accurate: 78.7% → 99.7%

Time: 19.6s

Alternatives: 39

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: 78.7% 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

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

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))
    (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin 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)) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
}

Julia

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

Derivation

1. Initial program 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. 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 89.0

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

3. Applied rewrites 89.0%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sin.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-sin.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. 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 89.0

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

3. Applied rewrites 89.0%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sin.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-sin.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in lambda1 around inf

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

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

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

   2. sin-diff N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sin.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.6× speedup

Math

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

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))
    (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin 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)) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(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)) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(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[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. 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 89.0

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

3. Applied rewrites 89.0%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 4: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.05:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3 \cdot \cos \phi_2}{t\_2}\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3 \cdot t\_0}{t\_1 - \left(\sin \phi_1 \cdot t\_0\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* phi2 phi2))))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (- t_1 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_3
         (fma
          (sin lambda1)
          (cos lambda2)
          (* (- (cos lambda1)) (sin lambda2)))))
   (if (<= phi2 -0.05)
     (atan2 (* t_3 (cos phi2)) t_2)
     (if (<= phi2 5e-47)
       (atan2
        (* t_3 t_0)
        (-
         t_1
         (*
          (* (sin phi1) t_0)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        t_2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - (0.5 * (phi2 * phi2));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = t_1 - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_3 = fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2)));
	double tmp;
	if (phi2 <= -0.05) {
		tmp = atan2((t_3 * cos(phi2)), t_2);
	} else if (phi2 <= 5e-47) {
		tmp = atan2((t_3 * t_0), (t_1 - ((sin(phi1) * t_0) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), t_2);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(phi2 * phi2)))
	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(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2)))
	tmp = 0.0
	if (phi2 <= -0.05)
		tmp = atan(Float64(t_3 * cos(phi2)), t_2);
	elseif (phi2 <= 5e-47)
		tmp = atan(Float64(t_3 * t_0), Float64(t_1 - Float64(Float64(sin(phi1) * t_0) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), t_2);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(phi2 * phi2), $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[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Cos[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.05], N[ArcTan[N[(t$95$3 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision], If[LessEqual[phi2, 5e-47], N[ArcTan[N[(t$95$3 * t$95$0), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(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]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.050000000000000003

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -0.050000000000000003 < phi2 < 5.00000000000000011e-47

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 55.4

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

   10. Applied rewrites 55.4%

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

   11. Taylor expanded in phi2 around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 55.9

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

   13. Applied rewrites 55.9%

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

   if 5.00000000000000011e-47 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

Alternative 5: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t\_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := 1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.05:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_2}\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot t\_3}{t\_1 - \left(\sin \phi_1 \cdot t\_3\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (- (* (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 (- 1.0 (* 0.5 (* phi2 phi2)))))
   (if (<= phi2 -0.05)
     (atan2
      (*
       (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
       (cos phi2))
      t_2)
     (if (<= phi2 5e-47)
       (atan2
        (* t_0 t_3)
        (-
         t_1
         (*
          (* (sin phi1) t_3)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2 (* t_0 (cos phi2)) t_2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (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 = 1.0 - (0.5 * (phi2 * phi2));
	double tmp;
	if (phi2 <= -0.05) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), t_2);
	} else if (phi2 <= 5e-47) {
		tmp = atan2((t_0 * t_3), (t_1 - ((sin(phi1) * t_3) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2((t_0 * cos(phi2)), t_2);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * 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 = Float64(1.0 - Float64(0.5 * Float64(phi2 * phi2)))
	tmp = 0.0
	if (phi2 <= -0.05)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), t_2);
	elseif (phi2 <= 5e-47)
		tmp = atan(Float64(t_0 * t_3), Float64(t_1 - Float64(Float64(sin(phi1) * t_3) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	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[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $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[(1.0 - N[(0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.05], 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$2], $MachinePrecision], If[LessEqual[phi2, 5e-47], 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[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $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 < -0.050000000000000003

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -0.050000000000000003 < phi2 < 5.00000000000000011e-47

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 55.4

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in phi2 around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 55.9

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

   11. Applied rewrites 55.9%

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

   if 5.00000000000000011e-47 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

Alternative 6: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_1}\\ \mathbf{if}\;\lambda_1 \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \mathsf{fma}\left(-\lambda_1, -\sin \lambda_2, \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2
         (atan2
          (*
           (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
           (cos phi2))
          (- t_0 (* t_1 (cos lambda1))))))
   (if (<= lambda1 -1.2e-15)
     t_2
     (if (<= lambda1 1.3e-13)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* t_1 (fma (- lambda1) (- (sin lambda2)) (cos 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((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	double tmp;
	if (lambda1 <= -1.2e-15) {
		tmp = t_2;
	} else if (lambda1 <= 1.3e-13) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * fma(-lambda1, -sin(lambda2), cos(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(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda1))))
	tmp = 0.0
	if (lambda1 <= -1.2e-15)
		tmp = t_2;
	elseif (lambda1 <= 1.3e-13)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * fma(Float64(-lambda1), Float64(-sin(lambda2)), cos(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[(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[(t$95$0 - N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -1.2e-15], t$95$2, If[LessEqual[lambda1, 1.3e-13], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[((-lambda1) * (-N[Sin[lambda2], $MachinePrecision]) + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.19999999999999997e-15 or 1.3e-13 < lambda1

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in lambda1 around inf

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

   6. Applied rewrites 79.6%

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

   if -1.19999999999999997e-15 < lambda1 < 1.3e-13

   1. Initial program 78.7%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 64.6

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

   4. Applied rewrites 64.6%

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

Alternative 7: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_3 := \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - t\_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -7.2 \cdot 10^{+16}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_2 \leq 0.0039:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 - t\_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2 (* (cos lambda1) (sin lambda2)))
        (t_3
         (atan2
          (* (- (* (sin lambda1) (cos lambda2)) t_2) (cos phi2))
          (- t_0 (* t_1 (cos lambda2))))))
   (if (<= lambda2 -7.2e+16)
     t_3
     (if (<= lambda2 0.0039)
       (atan2
        (* (- (sin lambda1) t_2) (cos phi2))
        (- t_0 (* t_1 (cos (- lambda1 lambda2)))))
       t_3))))

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) * sin(lambda2);
	double t_3 = atan2((((sin(lambda1) * cos(lambda2)) - t_2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	double tmp;
	if (lambda2 <= -7.2e+16) {
		tmp = t_3;
	} else if (lambda2 <= 0.0039) {
		tmp = atan2(((sin(lambda1) - t_2) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(phi1) * cos(phi2)
    t_2 = cos(lambda1) * sin(lambda2)
    t_3 = atan2((((sin(lambda1) * cos(lambda2)) - t_2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))))
    if (lambda2 <= (-7.2d+16)) then
        tmp = t_3
    else if (lambda2 <= 0.0039d0) then
        tmp = atan2(((sin(lambda1) - t_2) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(phi1) * Math.cos(phi2);
	double t_2 = Math.cos(lambda1) * Math.sin(lambda2);
	double t_3 = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - t_2) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda2))));
	double tmp;
	if (lambda2 <= -7.2e+16) {
		tmp = t_3;
	} else if (lambda2 <= 0.0039) {
		tmp = Math.atan2(((Math.sin(lambda1) - t_2) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin(phi1) * math.cos(phi2)
	t_2 = math.cos(lambda1) * math.sin(lambda2)
	t_3 = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - t_2) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda2))))
	tmp = 0
	if lambda2 <= -7.2e+16:
		tmp = t_3
	elif lambda2 <= 0.0039:
		tmp = math.atan2(((math.sin(lambda1) - t_2) * math.cos(phi2)), (t_0 - (t_1 * math.cos((lambda1 - lambda2)))))
	else:
		tmp = t_3
	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 = Float64(cos(lambda1) * sin(lambda2))
	t_3 = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - t_2) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))))
	tmp = 0.0
	if (lambda2 <= -7.2e+16)
		tmp = t_3;
	elseif (lambda2 <= 0.0039)
		tmp = atan(Float64(Float64(sin(lambda1) - t_2) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin(phi1) * cos(phi2);
	t_2 = cos(lambda1) * sin(lambda2);
	t_3 = atan2((((sin(lambda1) * cos(lambda2)) - t_2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	tmp = 0.0;
	if (lambda2 <= -7.2e+16)
		tmp = t_3;
	elseif (lambda2 <= 0.0039)
		tmp = atan2(((sin(lambda1) - t_2) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -7.2e+16], t$95$3, If[LessEqual[lambda2, 0.0039], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] - t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.2e16 or 0.0038999999999999998 < lambda2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in lambda1 around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 79.1

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

   6. Applied rewrites 79.1%

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

   if -7.2e16 < lambda2 < 0.0038999999999999998

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lift-sin.f64 80.6

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

   6. Applied rewrites 80.6%

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

Alternative 8: 88.8% accurate, 0.7× speedup

Math

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

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(Float64(-cos(lambda1)) * 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 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. 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 89.0

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

3. Applied rewrites 89.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sin.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-sin.f64 89.0

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

5. Applied rewrites 89.0%

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

Alternative 9: 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 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. 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 89.0

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

3. Applied rewrites 89.0%

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

Alternative 10: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\left(\sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_0}\\ \mathbf{if}\;\phi_1 \leq -2.45 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + \left(-\left(\phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1
         (atan2
          (* (- (sin lambda1) (* (cos lambda1) (sin lambda2))) (cos phi2))
          (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) t_0)))))
   (if (<= phi1 -2.45e-8)
     t_1
     (if (<= phi1 7e-5)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (cos phi2))
        (+ (sin phi2) (- (* (* phi1 (cos phi2)) t_0))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = atan2(((sin(lambda1) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_0)));
	double tmp;
	if (phi1 <= -2.45e-8) {
		tmp = t_1;
	} else if (phi1 <= 7e-5) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) + -((phi1 * cos(phi2)) * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(Float64(sin(lambda1) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * t_0)))
	tmp = 0.0
	if (phi1 <= -2.45e-8)
		tmp = t_1;
	elseif (phi1 <= 7e-5)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) + Float64(-Float64(Float64(phi1 * cos(phi2)) * t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[(N[Sin[lambda1], $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] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.45e-8], t$95$1, If[LessEqual[phi1, 7e-5], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Cos[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] + (-N[(N[(phi1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.4500000000000001e-8 or 6.9999999999999994e-5 < phi1

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lift-sin.f64 80.6

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

   6. Applied rewrites 80.6%

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

   if -2.4500000000000001e-8 < phi1 < 6.9999999999999994e-5

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. cos-diff-rev N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 57.4%

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

Alternative 11: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_0}\\ \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 + \left(-\left(\phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1
         (atan2
          (* (sin (- lambda1 lambda2)) (cos phi2))
          (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) t_0)))))
   (if (<= phi1 -3.6e-8)
     t_1
     (if (<= phi1 7e-5)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (cos phi2))
        (+ (sin phi2) (- (* (* phi1 (cos phi2)) t_0))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_0)));
	double tmp;
	if (phi1 <= -3.6e-8) {
		tmp = t_1;
	} else if (phi1 <= 7e-5) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) + -((phi1 * cos(phi2)) * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * t_0)))
	tmp = 0.0
	if (phi1 <= -3.6e-8)
		tmp = t_1;
	elseif (phi1 <= 7e-5)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) + Float64(-Float64(Float64(phi1 * cos(phi2)) * t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[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] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.6e-8], t$95$1, If[LessEqual[phi1, 7e-5], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Cos[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] + (-N[(N[(phi1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.59999999999999981e-8 or 6.9999999999999994e-5 < phi1

   1. Initial program 78.7%

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

   if -3.59999999999999981e-8 < phi1 < 6.9999999999999994e-5

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. cos-diff-rev N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 57.4%

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

Alternative 12: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t\_0}\\ \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot t\_0\right) + \sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1
         (atan2
          (* (sin (- lambda1 lambda2)) (cos phi2))
          (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) t_0)))))
   (if (<= phi1 -3.6e-8)
     t_1
     (if (<= phi1 7e-5)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (+ (- (* (* (cos phi2) phi1) t_0)) (sin phi2)))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_0)));
	double tmp;
	if (phi1 <= -3.6e-8) {
		tmp = t_1;
	} else if (phi1 <= 7e-5) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_0)))
    if (phi1 <= (-3.6d-8)) then
        tmp = t_1
    else if (phi1 <= 7d-5) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (-((cos(phi2) * phi1) * t_0) + sin(phi2)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * t_0)));
	double tmp;
	if (phi1 <= -3.6e-8) {
		tmp = t_1;
	} else if (phi1 <= 7e-5) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (-((Math.cos(phi2) * phi1) * t_0) + Math.sin(phi2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * t_0)))
	tmp = 0
	if phi1 <= -3.6e-8:
		tmp = t_1
	elif phi1 <= 7e-5:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (-((math.cos(phi2) * phi1) * t_0) + math.sin(phi2)))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * t_0)))
	tmp = 0.0
	if (phi1 <= -3.6e-8)
		tmp = t_1;
	elseif (phi1 <= 7e-5)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(-Float64(Float64(cos(phi2) * phi1) * t_0)) + sin(phi2)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * t_0)));
	tmp = 0.0;
	if (phi1 <= -3.6e-8)
		tmp = t_1;
	elseif (phi1 <= 7e-5)
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Sin[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] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.6e-8], t$95$1, If[LessEqual[phi1, 7e-5], 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[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision] * t$95$0), $MachinePrecision]) + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.59999999999999981e-8 or 6.9999999999999994e-5 < phi1

   1. Initial program 78.7%

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

   if -3.59999999999999981e-8 < phi1 < 6.9999999999999994e-5

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sin.f64 57.4

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

   6. Applied rewrites 57.4%

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

Alternative 13: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -3.7e-27)
     t_0
     (if (<= phi1 1.1e-46)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (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 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -3.7e-27) {
		tmp = t_0;
	} else if (phi1 <= 1.1e-46) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), 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 <= -3.7e-27)
		tmp = t_0;
	elseif (phi1 <= 1.1e-46)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(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, -3.7e-27], t$95$0, If[LessEqual[phi1, 1.1e-46], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.70000000000000029e-27 or 1.1e-46 < phi1

   1. Initial program 78.7%

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

   if -3.70000000000000029e-27 < phi1 < 1.1e-46

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

Alternative 14: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00075:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_1 \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (<= lambda1 -0.00075)
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (* (sin phi1) (cos phi2)) (cos lambda1))))
     (if (<= lambda1 1.15e-8)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2))))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (cos phi2))
        (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -0.00075) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	} else if (lambda1 <= 1.15e-8) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.00075)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda1 <= 1.15e-8)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * 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]}, If[LessEqual[lambda1, -0.00075], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 1.15e-8], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -7.5000000000000002e-4

   1. Initial program 78.7%

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

   2. 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 47.9%

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

   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 47.9%

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

   if -7.5000000000000002e-4 < lambda1 < 1.15e-8

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   if 1.15e-8 < lambda1

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

Alternative 15: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{+185}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_2 \leq -0.000125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{-199}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_2 - t\_0}\\ \mathbf{elif}\;\lambda_2 \leq 0.013:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - t\_0}\\ \mathbf{elif}\;\lambda_2 \leq 1.3 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))
        (t_1
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (- (cos lambda1)) (sin lambda2)))
           (cos phi2))
          (sin phi2)))
        (t_2 (* (cos phi1) (sin phi2)))
        (t_3
         (atan2
          (* (sin (- lambda2)) (cos phi2))
          (- t_2 (* (* (cos lambda2) (sin phi1)) (cos phi2))))))
   (if (<= lambda2 -8.5e+185)
     t_3
     (if (<= lambda2 -0.000125)
       t_1
       (if (<= lambda2 1.65e-199)
         (atan2 (* (sin lambda1) (cos phi2)) (- t_2 t_0))
         (if (<= lambda2 0.013)
           (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (sin phi2) t_0))
           (if (<= lambda2 1.3e+170) t_1 t_3)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2));
	double t_1 = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	double t_2 = cos(phi1) * sin(phi2);
	double t_3 = atan2((sin(-lambda2) * cos(phi2)), (t_2 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	double tmp;
	if (lambda2 <= -8.5e+185) {
		tmp = t_3;
	} else if (lambda2 <= -0.000125) {
		tmp = t_1;
	} else if (lambda2 <= 1.65e-199) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_2 - t_0));
	} else if (lambda2 <= 0.013) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - t_0));
	} else if (lambda2 <= 1.3e+170) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))
	t_1 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	t_3 = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_2 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))))
	tmp = 0.0
	if (lambda2 <= -8.5e+185)
		tmp = t_3;
	elseif (lambda2 <= -0.000125)
		tmp = t_1;
	elseif (lambda2 <= 1.65e-199)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_2 - t_0));
	elseif (lambda2 <= 0.013)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(sin(phi2) - t_0));
	elseif (lambda2 <= 1.3e+170)
		tmp = t_1;
	else
		tmp = t_3;
	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[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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -8.5e+185], t$95$3, If[LessEqual[lambda2, -0.000125], t$95$1, If[LessEqual[lambda2, 1.65e-199], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.013], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 1.3e+170], t$95$1, t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -8.50000000000000013e185 or 1.2999999999999999e170 < lambda2

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 47.2

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

   7. Applied rewrites 47.2%

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

   if -8.50000000000000013e185 < lambda2 < -1.25e-4 or 0.0129999999999999994 < lambda2 < 1.2999999999999999e170

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

   if -1.25e-4 < lambda2 < 1.6500000000000001e-199

   1. Initial program 78.7%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 47.9

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

   4. Applied rewrites 47.9%

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

   if 1.6500000000000001e-199 < lambda2 < 0.0129999999999999994

   1. Initial program 78.7%

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

   2. 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 65.7

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

   4. Applied rewrites 65.7%

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

Alternative 16: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ t_2 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_3 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_2 \leq -0.000125:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{-199}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - t\_3 \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_2 \leq 0.013:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - t\_3 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.3 \cdot 10^{+170}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (atan2
          (* (sin (- lambda2)) (cos phi2))
          (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2)))))
        (t_2
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (- (cos lambda1)) (sin lambda2)))
           (cos phi2))
          (sin phi2)))
        (t_3 (* (sin phi1) (cos phi2))))
   (if (<= lambda2 -8.5e+185)
     t_1
     (if (<= lambda2 -0.000125)
       t_2
       (if (<= lambda2 1.65e-199)
         (atan2 (* (sin lambda1) (cos phi2)) (- t_0 (* t_3 (cos lambda1))))
         (if (<= lambda2 0.013)
           (atan2
            (* (sin (- lambda1 lambda2)) (cos phi2))
            (- (sin phi2) (* t_3 (cos (- lambda1 lambda2)))))
           (if (<= lambda2 1.3e+170) t_2 t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	double t_2 = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	double t_3 = sin(phi1) * cos(phi2);
	double tmp;
	if (lambda2 <= -8.5e+185) {
		tmp = t_1;
	} else if (lambda2 <= -0.000125) {
		tmp = t_2;
	} else if (lambda2 <= 1.65e-199) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_3 * cos(lambda1))));
	} else if (lambda2 <= 0.013) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - (t_3 * cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 1.3e+170) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))))
	t_2 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2))
	t_3 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -8.5e+185)
		tmp = t_1;
	elseif (lambda2 <= -0.000125)
		tmp = t_2;
	elseif (lambda2 <= 1.65e-199)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(t_3 * cos(lambda1))));
	elseif (lambda2 <= 0.013)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(sin(phi2) - Float64(t_3 * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda2 <= 1.3e+170)
		tmp = t_2;
	else
		tmp = 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[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]}, 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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -8.5e+185], t$95$1, If[LessEqual[lambda2, -0.000125], t$95$2, If[LessEqual[lambda2, 1.65e-199], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$3 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.013], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$3 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 1.3e+170], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -8.50000000000000013e185 or 1.2999999999999999e170 < lambda2

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 47.2

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

   7. Applied rewrites 47.2%

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

   if -8.50000000000000013e185 < lambda2 < -1.25e-4 or 0.0129999999999999994 < lambda2 < 1.2999999999999999e170

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

   if -1.25e-4 < lambda2 < 1.6500000000000001e-199

   1. Initial program 78.7%

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

   2. 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 47.9%

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

   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 47.9%

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

   if 1.6500000000000001e-199 < lambda2 < 0.0129999999999999994

   1. Initial program 78.7%

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

   2. 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 65.7

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

   4. Applied rewrites 65.7%

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

Alternative 17: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.000125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{-199}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_1 \cdot \cos \lambda_1}\\ \mathbf{elif}\;\lambda_2 \leq 0.013:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (- (cos lambda1)) (sin lambda2)))
           (cos phi2))
          (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2))))
   (if (<= lambda2 -0.000125)
     t_0
     (if (<= lambda2 1.65e-199)
       (atan2
        (* (sin lambda1) (cos phi2))
        (- (* (cos phi1) (sin phi2)) (* t_1 (cos lambda1))))
       (if (<= lambda2 0.013)
         (atan2
          (* (sin (- lambda1 lambda2)) (cos phi2))
          (- (sin phi2) (* t_1 (cos (- lambda1 lambda2)))))
         t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	double t_1 = sin(phi1) * cos(phi2);
	double tmp;
	if (lambda2 <= -0.000125) {
		tmp = t_0;
	} else if (lambda2 <= 1.65e-199) {
		tmp = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (t_1 * cos(lambda1))));
	} else if (lambda2 <= 0.013) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (sin(phi2) - (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -0.000125)
		tmp = t_0;
	elseif (lambda2 <= 1.65e-199)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_1 * cos(lambda1))));
	elseif (lambda2 <= 0.013)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(sin(phi2) - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[(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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -0.000125], t$95$0, If[LessEqual[lambda2, 1.65e-199], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.013], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.25e-4 or 0.0129999999999999994 < lambda2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

   if -1.25e-4 < lambda2 < 1.6500000000000001e-199

   1. Initial program 78.7%

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

   2. 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 47.9%

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

   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 47.9%

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

   if 1.6500000000000001e-199 < lambda2 < 0.0129999999999999994

   1. Initial program 78.7%

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

   2. 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 65.7

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

   4. Applied rewrites 65.7%

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

Alternative 18: 70.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 8 \cdot 10^{+87}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (- lambda1 lambda2) -0.002)
     (atan2 t_1 (- t_0 (* (sin phi1) (cos (- lambda1 lambda2)))))
     (if (<= (- lambda1 lambda2) 8e+87)
       (atan2 t_1 (- t_0 (* (sin phi1) (cos phi2))))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (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)) * cos(phi2);
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = atan2(t_1, (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else if ((lambda1 - lambda2) <= 8e+87) {
		tmp = atan2(t_1, (t_0 - (sin(phi1) * cos(phi2))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -0.002)
		tmp = atan(t_1, Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	elseif (Float64(lambda1 - lambda2) <= 8e+87)
		tmp = atan(t_1, Float64(t_0 - Float64(sin(phi1) * cos(phi2))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * 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[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -0.002], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 8e+87], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $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] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -2e-3

   1. Initial program 78.7%

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

   2. 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 66.1

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

   4. Applied rewrites 66.1%

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

   if -2e-3 < (-.f64 lambda1 lambda2) < 7.9999999999999997e87

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 59.5

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

   7. Applied rewrites 59.5%

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

   if 7.9999999999999997e87 < (-.f64 lambda1 lambda2)

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

Alternative 19: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 8 \cdot 10^{+87}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (- lambda1 lambda2) -0.002)
     (atan2 t_1 (- (sin phi2) (* t_0 (cos (- lambda1 lambda2)))))
     (if (<= (- lambda1 lambda2) 8e+87)
       (atan2 t_1 (- (* (cos phi1) (sin phi2)) t_0))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
         (cos phi2))
        (sin phi2))))))

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 ((lambda1 - lambda2) <= -0.002) {
		tmp = atan2(t_1, (sin(phi2) - (t_0 * cos((lambda1 - lambda2)))));
	} else if ((lambda1 - lambda2) <= 8e+87) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - t_0));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	}
	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 (Float64(lambda1 - lambda2) <= -0.002)
		tmp = atan(t_1, Float64(sin(phi2) - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (Float64(lambda1 - lambda2) <= 8e+87)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - t_0));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2));
	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[N[(lambda1 - lambda2), $MachinePrecision], -0.002], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 8e+87], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $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] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -2e-3

   1. Initial program 78.7%

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

   2. 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 65.7

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

   4. Applied rewrites 65.7%

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

   if -2e-3 < (-.f64 lambda1 lambda2) < 7.9999999999999997e87

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 59.5

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

   7. Applied rewrites 59.5%

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

   if 7.9999999999999997e87 < (-.f64 lambda1 lambda2)

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

Alternative 20: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \phi_2}\\ t_2 := \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ \mathbf{if}\;\phi_2 \leq -21000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot t\_2}{\cos \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right) - \left(\sin \phi_1 \cdot t\_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1
         (atan2
          (* t_0 (cos phi2))
          (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos phi2)))))
        (t_2
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0)))
   (if (<= phi2 -21000.0)
     t_1
     (if (<= phi2 1.05e-36)
       (atan2
        (* t_0 t_2)
        (-
         (*
          (cos phi1)
          (*
           (fma
            (-
             (*
              (fma (* phi2 phi2) -0.0001984126984126984 0.008333333333333333)
              (* phi2 phi2))
             0.16666666666666666)
            (* phi2 phi2)
            1.0)
           phi2))
         (* (* (sin phi1) t_2) (cos (- lambda1 lambda2)))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos(phi2))));
	double t_2 = fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0);
	double tmp;
	if (phi2 <= -21000.0) {
		tmp = t_1;
	} else if (phi2 <= 1.05e-36) {
		tmp = atan2((t_0 * t_2), ((cos(phi1) * (fma(((fma((phi2 * phi2), -0.0001984126984126984, 0.008333333333333333) * (phi2 * phi2)) - 0.16666666666666666), (phi2 * phi2), 1.0) * phi2)) - ((sin(phi1) * t_2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(phi2))))
	t_2 = fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	tmp = 0.0
	if (phi2 <= -21000.0)
		tmp = t_1;
	elseif (phi2 <= 1.05e-36)
		tmp = atan(Float64(t_0 * t_2), Float64(Float64(cos(phi1) * Float64(fma(Float64(Float64(fma(Float64(phi2 * phi2), -0.0001984126984126984, 0.008333333333333333) * Float64(phi2 * phi2)) - 0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2)) - Float64(Float64(sin(phi1) * t_2) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[phi2, -21000.0], t$95$1, If[LessEqual[phi2, 1.05e-36], N[ArcTan[N[(t$95$0 * t$95$2), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -21000 or 1.04999999999999995e-36 < phi2

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 59.5

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

   7. Applied rewrites 59.5%

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

   if -21000 < phi2 < 1.04999999999999995e-36

   1. Initial program 78.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)\right)} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 43.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right)} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{if}\;\phi_2 \leq -21000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot 1}{t\_0 - \left(\sin \phi_1 \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (atan2 (* t_1 (cos phi2)) (- t_0 (* (sin phi1) (cos phi2))))))
   (if (<= phi2 -21000.0)
     t_2
     (if (<= phi2 1.05e-36)
       (atan2
        (* t_1 1.0)
        (- t_0 (* (* (sin phi1) 1.0) (cos (- lambda1 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((lambda1 - lambda2));
	double t_2 = atan2((t_1 * cos(phi2)), (t_0 - (sin(phi1) * cos(phi2))));
	double tmp;
	if (phi2 <= -21000.0) {
		tmp = t_2;
	} else if (phi2 <= 1.05e-36) {
		tmp = atan2((t_1 * 1.0), (t_0 - ((sin(phi1) * 1.0) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin((lambda1 - lambda2))
    t_2 = atan2((t_1 * cos(phi2)), (t_0 - (sin(phi1) * cos(phi2))))
    if (phi2 <= (-21000.0d0)) then
        tmp = t_2
    else if (phi2 <= 1.05d-36) then
        tmp = atan2((t_1 * 1.0d0), (t_0 - ((sin(phi1) * 1.0d0) * cos((lambda1 - lambda2)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin((lambda1 - lambda2));
	double t_2 = Math.atan2((t_1 * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * Math.cos(phi2))));
	double tmp;
	if (phi2 <= -21000.0) {
		tmp = t_2;
	} else if (phi2 <= 1.05e-36) {
		tmp = Math.atan2((t_1 * 1.0), (t_0 - ((Math.sin(phi1) * 1.0) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = math.atan2((t_1 * math.cos(phi2)), (t_0 - (math.sin(phi1) * math.cos(phi2))))
	tmp = 0
	if phi2 <= -21000.0:
		tmp = t_2
	elif phi2 <= 1.05e-36:
		tmp = math.atan2((t_1 * 1.0), (t_0 - ((math.sin(phi1) * 1.0) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = atan(Float64(t_1 * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(phi2))))
	tmp = 0.0
	if (phi2 <= -21000.0)
		tmp = t_2;
	elseif (phi2 <= 1.05e-36)
		tmp = atan(Float64(t_1 * 1.0), Float64(t_0 - Float64(Float64(sin(phi1) * 1.0) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin((lambda1 - lambda2));
	t_2 = atan2((t_1 * cos(phi2)), (t_0 - (sin(phi1) * cos(phi2))));
	tmp = 0.0;
	if (phi2 <= -21000.0)
		tmp = t_2;
	elseif (phi2 <= 1.05e-36)
		tmp = atan2((t_1 * 1.0), (t_0 - ((sin(phi1) * 1.0) * cos((lambda1 - lambda2)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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, -21000.0], t$95$2, If[LessEqual[phi2, 1.05e-36], N[ArcTan[N[(t$95$1 * 1.0), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -21000 or 1.04999999999999995e-36 < phi2

   1. Initial program 78.7%

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

   2. 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 68.9

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

   4. Applied rewrites 68.9%

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

   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 59.5

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

   7. Applied rewrites 59.5%

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

   if -21000 < phi2 < 1.04999999999999995e-36

   1. Initial program 78.7%

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

   2. Taylor expanded in phi2 around 0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 49.0%

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

Alternative 22: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -2.15 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (- (cos lambda1)) (sin lambda2)))
           (cos phi2))
          (sin phi2))))
   (if (<= phi2 -2.15e-11)
     t_0
     (if (<= phi2 1.05e-36)
       (atan2
        (* (sin (- lambda1 lambda2)) 1.0)
        (-
         (* (cos phi1) (sin phi2))
         (* (* (sin phi1) 1.0) (cos (- lambda1 lambda2)))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -2.15e-11) {
		tmp = t_0;
	} else if (phi2 <= 1.05e-36) {
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * 1.0) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.15e-11)
		tmp = t_0;
	elseif (phi2 <= 1.05e-36)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * 1.0), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * 1.0) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[(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[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -2.15e-11], t$95$0, If[LessEqual[phi2, 1.05e-36], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.15000000000000001e-11 or 1.04999999999999995e-36 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. 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 89.0

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

   3. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-diff N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi1 around 0

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

      1. cos-diff-rev N/A

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

      2. lift-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

   if -2.15000000000000001e-11 < phi2 < 1.04999999999999995e-36

   1. Initial program 78.7%

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

   2. Taylor expanded in phi2 around 0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 49.0%

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

Alternative 23: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-t\_0 \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot t\_0\right) + \sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot 1}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot 1\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi1 -8.5e+17)
     (atan2 t_2 (- (* t_0 (sin phi1))))
     (if (<= phi1 1.55e-11)
       (atan2 t_2 (+ (- (* (* (cos phi2) phi1) t_0)) (sin phi2)))
       (atan2
        (* t_1 1.0)
        (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) 1.0) 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));
	double t_2 = t_1 * cos(phi2);
	double tmp;
	if (phi1 <= -8.5e+17) {
		tmp = atan2(t_2, -(t_0 * sin(phi1)));
	} else if (phi1 <= 1.55e-11) {
		tmp = atan2(t_2, (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	} else {
		tmp = atan2((t_1 * 1.0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * 1.0) * 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 = cos((lambda1 - lambda2))
    t_1 = sin((lambda1 - lambda2))
    t_2 = t_1 * cos(phi2)
    if (phi1 <= (-8.5d+17)) then
        tmp = atan2(t_2, -(t_0 * sin(phi1)))
    else if (phi1 <= 1.55d-11) then
        tmp = atan2(t_2, (-((cos(phi2) * phi1) * t_0) + sin(phi2)))
    else
        tmp = atan2((t_1 * 1.0d0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * 1.0d0) * 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));
	double t_2 = t_1 * Math.cos(phi2);
	double tmp;
	if (phi1 <= -8.5e+17) {
		tmp = Math.atan2(t_2, -(t_0 * Math.sin(phi1)));
	} else if (phi1 <= 1.55e-11) {
		tmp = Math.atan2(t_2, (-((Math.cos(phi2) * phi1) * t_0) + Math.sin(phi2)));
	} else {
		tmp = Math.atan2((t_1 * 1.0), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * 1.0) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = t_1 * math.cos(phi2)
	tmp = 0
	if phi1 <= -8.5e+17:
		tmp = math.atan2(t_2, -(t_0 * math.sin(phi1)))
	elif phi1 <= 1.55e-11:
		tmp = math.atan2(t_2, (-((math.cos(phi2) * phi1) * t_0) + math.sin(phi2)))
	else:
		tmp = math.atan2((t_1 * 1.0), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * 1.0) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi1 <= -8.5e+17)
		tmp = atan(t_2, Float64(-Float64(t_0 * sin(phi1))));
	elseif (phi1 <= 1.55e-11)
		tmp = atan(t_2, Float64(Float64(-Float64(Float64(cos(phi2) * phi1) * t_0)) + sin(phi2)));
	else
		tmp = atan(Float64(t_1 * 1.0), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * 1.0) * 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));
	t_2 = t_1 * cos(phi2);
	tmp = 0.0;
	if (phi1 <= -8.5e+17)
		tmp = atan2(t_2, -(t_0 * sin(phi1)));
	elseif (phi1 <= 1.55e-11)
		tmp = atan2(t_2, (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	else
		tmp = atan2((t_1 * 1.0), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * 1.0) * 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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.5e+17], N[ArcTan[t$95$2 / (-N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])], $MachinePrecision], If[LessEqual[phi1, 1.55e-11], N[ArcTan[t$95$2 / N[((-N[(N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision] * t$95$0), $MachinePrecision]) + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$1 * 1.0), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.5e17

   1. Initial program 78.7%

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

   2. 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. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 48.3

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

   4. Applied rewrites 48.3%

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

   if -8.5e17 < phi1 < 1.55000000000000014e-11

   1. Initial program 78.7%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sin.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if 1.55000000000000014e-11 < phi1

   1. Initial program 78.7%

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

   2. Taylor expanded in phi2 around 0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 49.0%

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

Alternative 24: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_1}{-t\_0 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 0.0048:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot t\_0\right) + \sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (atan2 t_1 (- (* t_0 (sin phi1))))))
   (if (<= phi1 -8.5e+17)
     t_2
     (if (<= phi1 0.0048)
       (atan2 t_1 (+ (- (* (* (cos phi2) phi1) t_0)) (sin phi2)))
       t_2))))

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 t_2 = atan2(t_1, -(t_0 * sin(phi1)));
	double tmp;
	if (phi1 <= -8.5e+17) {
		tmp = t_2;
	} else if (phi1 <= 0.0048) {
		tmp = atan2(t_1, (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin((lambda1 - lambda2)) * cos(phi2)
    t_2 = atan2(t_1, -(t_0 * sin(phi1)))
    if (phi1 <= (-8.5d+17)) then
        tmp = t_2
    else if (phi1 <= 0.0048d0) then
        tmp = atan2(t_1, (-((cos(phi2) * phi1) * t_0) + sin(phi2)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double t_2 = Math.atan2(t_1, -(t_0 * Math.sin(phi1)));
	double tmp;
	if (phi1 <= -8.5e+17) {
		tmp = t_2;
	} else if (phi1 <= 0.0048) {
		tmp = Math.atan2(t_1, (-((Math.cos(phi2) * phi1) * t_0) + Math.sin(phi2)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	t_2 = math.atan2(t_1, -(t_0 * math.sin(phi1)))
	tmp = 0
	if phi1 <= -8.5e+17:
		tmp = t_2
	elif phi1 <= 0.0048:
		tmp = math.atan2(t_1, (-((math.cos(phi2) * phi1) * t_0) + math.sin(phi2)))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	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 <= -8.5e+17)
		tmp = t_2;
	elseif (phi1 <= 0.0048)
		tmp = atan(t_1, Float64(Float64(-Float64(Float64(cos(phi2) * phi1) * t_0)) + sin(phi2)));
	else
		tmp = t_2;
	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);
	t_2 = atan2(t_1, -(t_0 * sin(phi1)));
	tmp = 0.0;
	if (phi1 <= -8.5e+17)
		tmp = t_2;
	elseif (phi1 <= 0.0048)
		tmp = atan2(t_1, (-((cos(phi2) * phi1) * t_0) + sin(phi2)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[ArcTan[t$95$1 / (-N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[phi1, -8.5e+17], t$95$2, If[LessEqual[phi1, 0.0048], N[ArcTan[t$95$1 / N[((-N[(N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision] * t$95$0), $MachinePrecision]) + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5e17 or 0.00479999999999999958 < phi1

   1. Initial program 78.7%

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

   2. 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. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 48.3

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

   4. Applied rewrites 48.3%

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

   if -8.5e17 < phi1 < 0.00479999999999999958

   1. Initial program 78.7%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \sin \phi_2} \]

      11. lift--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \sin \phi_2} \]

      12. lift-sin.f64 48.0

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \sin \phi_2} \]

   4. Applied rewrites 48.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(-\left(\cos \phi_2 \cdot \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \sin \phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -1.5 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 8200:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\sin \phi_2 - \left(\sin \phi_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma -0.5 (* phi2 phi2) 1.0))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (atan2 (* t_1 (cos phi2)) (sin phi2))))
   (if (<= phi2 -1.5e-10)
     t_2
     (if (<= phi2 8200.0)
       (atan2
        (* t_1 t_0)
        (- (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(-0.5, (phi2 * phi2), 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = atan2((t_1 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -1.5e-10) {
		tmp = t_2;
	} else if (phi2 <= 8200.0) {
		tmp = atan2((t_1 * t_0), (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(-0.5, Float64(phi2 * phi2), 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = atan(Float64(t_1 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.5e-10)
		tmp = t_2;
	elseif (phi2 <= 8200.0)
		tmp = atan(Float64(t_1 * t_0), Float64(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[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 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, -1.5e-10], t$95$2, If[LessEqual[phi2, 8200.0], N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Sin[phi2], $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 < -1.5e-10 or 8200 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -1.5e-10 < phi2 < 8200

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.9

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 45.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   9. Step-by-step derivation

      1. lift-sin.f64 45.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 45.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2} \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(-0.5, \color{blue}{\phi_2} \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2} \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   15. Step-by-step derivation

   16. Applied rewrites 46.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\phi_2} \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 63.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -21000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\phi_2 \cdot \left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - \left(\sin \phi_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (atan2 (* t_1 (cos phi2)) (sin phi2))))
   (if (<= phi2 -21000.0)
     t_2
     (if (<= phi2 1.05e-36)
       (atan2
        (* t_1 t_0)
        (-
         (* phi2 (- 1.0 (* 0.16666666666666666 (* phi2 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) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = atan2((t_1 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -21000.0) {
		tmp = t_2;
	} else if (phi2 <= 1.05e-36) {
		tmp = atan2((t_1 * t_0), ((phi2 * (1.0 - (0.16666666666666666 * (phi2 * 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(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = atan(Float64(t_1 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -21000.0)
		tmp = t_2;
	elseif (phi2 <= 1.05e-36)
		tmp = atan(Float64(t_1 * t_0), Float64(Float64(phi2 * Float64(1.0 - Float64(0.16666666666666666 * Float64(phi2 * 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[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 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, -21000.0], t$95$2, If[LessEqual[phi2, 1.05e-36], N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(phi2 * N[(1.0 - N[(0.16666666666666666 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $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 < -21000 or 1.04999999999999995e-36 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -21000 < phi2 < 1.04999999999999995e-36

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.9

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 45.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   9. Step-by-step derivation

      1. lift-sin.f64 45.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 45.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \color{blue}{{\phi_2}^{2}}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \color{blue}{{\phi_2}^{2}}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot {\phi_2}^{2}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot {\phi_2}^{\color{blue}{2}}\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       6. pow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot \left(\phi_2 \cdot \phi_2\right)\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

       7. lift-*.f64 44.1

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   13. Applied rewrites 44.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right)} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -21000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{\sin \phi_2 - \left(\sin \phi_1 \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -21000.0)
     t_1
     (if (<= phi2 1.05e-36)
       (atan2
        (* t_0 1.0)
        (- (sin phi2) (* (* (sin phi1) 1.0) (cos (- lambda1 lambda2)))))
       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 <= -21000.0) {
		tmp = t_1;
	} else if (phi2 <= 1.05e-36) {
		tmp = atan2((t_0 * 1.0), (sin(phi2) - ((sin(phi1) * 1.0) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = atan2((t_0 * cos(phi2)), sin(phi2))
    if (phi2 <= (-21000.0d0)) then
        tmp = t_1
    else if (phi2 <= 1.05d-36) then
        tmp = atan2((t_0 * 1.0d0), (sin(phi2) - ((sin(phi1) * 1.0d0) * cos((lambda1 - lambda2)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2((t_0 * Math.cos(phi2)), Math.sin(phi2));
	double tmp;
	if (phi2 <= -21000.0) {
		tmp = t_1;
	} else if (phi2 <= 1.05e-36) {
		tmp = Math.atan2((t_0 * 1.0), (Math.sin(phi2) - ((Math.sin(phi1) * 1.0) * Math.cos((lambda1 - lambda2)))));
	} 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 <= -21000.0:
		tmp = t_1
	elif phi2 <= 1.05e-36:
		tmp = math.atan2((t_0 * 1.0), (math.sin(phi2) - ((math.sin(phi1) * 1.0) * math.cos((lambda1 - lambda2)))))
	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 <= -21000.0)
		tmp = t_1;
	elseif (phi2 <= 1.05e-36)
		tmp = atan(Float64(t_0 * 1.0), Float64(sin(phi2) - Float64(Float64(sin(phi1) * 1.0) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -21000.0)
		tmp = t_1;
	elseif (phi2 <= 1.05e-36)
		tmp = atan2((t_0 * 1.0), (sin(phi2) - ((sin(phi1) * 1.0) * cos((lambda1 - lambda2)))));
	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, -21000.0], t$95$1, If[LessEqual[phi2, 1.05e-36], N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -21000 or 1.04999999999999995e-36 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -21000 < phi2 < 1.04999999999999995e-36

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. lower-*.f64 45.9

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 45.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   9. Step-by-step derivation

      1. lift-sin.f64 45.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 45.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)}{\color{blue}{\sin \phi_2} - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{24} - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\sin \phi_2 - \left(\sin \phi_1 \cdot \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\sin \phi_2 - \left(\sin \phi_1 \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   15. Step-by-step derivation

   16. Applied rewrites 48.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\sin \phi_2 - \left(\sin \phi_1 \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -3 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 10^{-33}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -3e-13)
     t_1
     (if (<= phi2 1e-33)
       (atan2 (* t_0 1.0) (- (* (cos (- lambda1 lambda2)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -3e-13) {
		tmp = t_1;
	} else if (phi2 <= 1e-33) {
		tmp = atan2((t_0 * 1.0), -(cos((lambda1 - lambda2)) * sin(phi1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = atan2((t_0 * cos(phi2)), sin(phi2))
    if (phi2 <= (-3d-13)) then
        tmp = t_1
    else if (phi2 <= 1d-33) then
        tmp = atan2((t_0 * 1.0d0), -(cos((lambda1 - lambda2)) * sin(phi1)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2((t_0 * Math.cos(phi2)), Math.sin(phi2));
	double tmp;
	if (phi2 <= -3e-13) {
		tmp = t_1;
	} else if (phi2 <= 1e-33) {
		tmp = Math.atan2((t_0 * 1.0), -(Math.cos((lambda1 - lambda2)) * Math.sin(phi1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.atan2((t_0 * math.cos(phi2)), math.sin(phi2))
	tmp = 0
	if phi2 <= -3e-13:
		tmp = t_1
	elif phi2 <= 1e-33:
		tmp = math.atan2((t_0 * 1.0), -(math.cos((lambda1 - lambda2)) * math.sin(phi1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -3e-13)
		tmp = t_1;
	elseif (phi2 <= 1e-33)
		tmp = atan(Float64(t_0 * 1.0), Float64(-Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -3e-13)
		tmp = t_1;
	elseif (phi2 <= 1e-33)
		tmp = atan2((t_0 * 1.0), -(cos((lambda1 - lambda2)) * sin(phi1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -3e-13], t$95$1, If[LessEqual[phi2, 1e-33], N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / (-N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.99999999999999984e-13 or 1.0000000000000001e-33 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -2.99999999999999984e-13 < phi2 < 1.0000000000000001e-33

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      5. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      6. lift-sin.f64 45.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   10. Applied rewrites 45.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\color{blue}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 48.6% 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 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. 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 48.6

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 48.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
5. Add Preprocessing

Alternative 30: 43.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -0.000125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (atan2 (* (sin (- lambda2)) (cos phi2)) (sin phi2))))
   (if (<= lambda2 -0.000125)
     t_0
     (if (<= lambda2 1.4e-53)
       (atan2 (* (sin lambda1) (cos phi2)) (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin(-lambda2) * cos(phi2)), sin(phi2));
	double tmp;
	if (lambda2 <= -0.000125) {
		tmp = t_0;
	} else if (lambda2 <= 1.4e-53) {
		tmp = atan2((sin(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(-lambda2) * cos(phi2)), sin(phi2))
    if (lambda2 <= (-0.000125d0)) then
        tmp = t_0
    else if (lambda2 <= 1.4d-53) then
        tmp = atan2((sin(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(-lambda2) * Math.cos(phi2)), Math.sin(phi2));
	double tmp;
	if (lambda2 <= -0.000125) {
		tmp = t_0;
	} else if (lambda2 <= 1.4e-53) {
		tmp = Math.atan2((Math.sin(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(-lambda2) * math.cos(phi2)), math.sin(phi2))
	tmp = 0
	if lambda2 <= -0.000125:
		tmp = t_0
	elif lambda2 <= 1.4e-53:
		tmp = math.atan2((math.sin(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(-lambda2)) * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (lambda2 <= -0.000125)
		tmp = t_0;
	elseif (lambda2 <= 1.4e-53)
		tmp = atan(Float64(sin(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(-lambda2) * cos(phi2)), sin(phi2));
	tmp = 0.0;
	if (lambda2 <= -0.000125)
		tmp = t_0;
	elseif (lambda2 <= 1.4e-53)
		tmp = atan2((sin(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[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -0.000125], t$95$0, If[LessEqual[lambda2, 1.4e-53], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.25e-4 or 1.39999999999999993e-53 < lambda2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in 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 32.3

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 32.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   if -1.25e-4 < lambda2 < 1.39999999999999993e-53

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in 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 32.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 37.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 38000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 38000.0)
   (atan2
    (* (sin (- lambda1 lambda2)) (cos phi2))
    (*
     (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 <= 38000.0) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (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 <= 38000.0)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(fma(Float64(Float64(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, 38000.0], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 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 < 38000

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
   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. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      8. lower-*.f64 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. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      11. pow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

      12. lift-*.f64 31.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]

   if 38000 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in 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 32.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot \cos \phi_2}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 36.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq 8200:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi2 8200.0)
     (atan2
      t_0
      (*
       (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 <= 8200.0) {
		tmp = atan2(t_0, (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 <= 8200.0)
		tmp = atan(t_0, Float64(fma(Float64(Float64(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, 8200.0], N[ArcTan[t$95$0 / N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 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 < 8200

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
   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. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      8. lower-*.f64 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. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2} \]

      11. pow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{1}{120} - \frac{1}{6}, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

      12. lift-*.f64 31.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.008333333333333333 - 0.16666666666666666, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\phi_2}} \]

   if 8200 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   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. pow2 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. lift-*.f64 31.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 31.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 36.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -21000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi2 -21000.0)
     (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 <= -21000.0) {
		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 <= -21000.0)
		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, -21000.0], 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 < -21000

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]

   if -21000 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   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. pow2 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. lift-*.f64 31.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2} \]

   7. Applied rewrites 31.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \color{blue}{\phi_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 34.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 2.8e+20)
     (atan2 (* t_0 (cos phi2)) phi2)
     (atan2 t_0 (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 2.8e+20) {
		tmp = atan2((t_0 * cos(phi2)), phi2);
	} else {
		tmp = atan2(t_0, 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) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (phi2 <= 2.8d+20) then
        tmp = atan2((t_0 * cos(phi2)), phi2)
    else
        tmp = atan2(t_0, 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.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 2.8e+20) {
		tmp = Math.atan2((t_0 * Math.cos(phi2)), phi2);
	} else {
		tmp = Math.atan2(t_0, Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 2.8e+20:
		tmp = math.atan2((t_0 * math.cos(phi2)), phi2)
	else:
		tmp = math.atan2(t_0, math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 2.8e+20)
		tmp = atan(Float64(t_0 * cos(phi2)), phi2);
	else
		tmp = atan(t_0, sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 2.8e+20)
		tmp = atan2((t_0 * cos(phi2)), phi2);
	else
		tmp = atan2(t_0, sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2.8e+20], N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.8e20

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2} \]

   if 2.8e20 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
   9. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

      2. lift--.f64 31.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

   10. Applied rewrites 31.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 35: 32.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -25000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -25000.0)
   (atan2 (* (cos phi2) (- (sin lambda2))) phi2)
   (atan2 (sin (- lambda1 lambda2)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -25000.0) {
		tmp = atan2((cos(phi2) * -sin(lambda2)), phi2);
	} else {
		tmp = atan2(sin((lambda1 - lambda2)), 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) :: tmp
    if (phi2 <= (-25000.0d0)) then
        tmp = atan2((cos(phi2) * -sin(lambda2)), phi2)
    else
        tmp = atan2(sin((lambda1 - lambda2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -25000.0) {
		tmp = Math.atan2((Math.cos(phi2) * -Math.sin(lambda2)), phi2);
	} else {
		tmp = Math.atan2(Math.sin((lambda1 - lambda2)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -25000.0:
		tmp = math.atan2((math.cos(phi2) * -math.sin(lambda2)), phi2)
	else:
		tmp = math.atan2(math.sin((lambda1 - lambda2)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -25000.0)
		tmp = atan(Float64(cos(phi2) * Float64(-sin(lambda2))), phi2);
	else
		tmp = atan(sin(Float64(lambda1 - lambda2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -25000.0)
		tmp = atan2((cos(phi2) * -sin(lambda2)), phi2);
	else
		tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -25000.0], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -25000

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   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 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   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 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}{\phi_2} \]

       2. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-1 \cdot \color{blue}{\sin \lambda_2}\right)}{\phi_2} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(-1 \cdot \sin \lambda_2\right)}}{\phi_2} \]

       4. lift-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\color{blue}{-1} \cdot \sin \lambda_2\right)}{\phi_2} \]

       5. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}{\phi_2} \]

       6. lower-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\phi_2} \]

       7. lift-sin.f64 23.0

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\phi_2} \]

   13. Applied rewrites 23.0%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}}{\phi_2} \]

   if -25000 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
   9. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

      2. lift--.f64 31.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

   10. Applied rewrites 31.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 36: 31.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot 1}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.55)
   (atan2 (* (sin lambda1) 1.0) phi2)
   (atan2
    (* (sin (- lambda1 lambda2)) 1.0)
    (*
     phi2
     (+
      1.0
      (*
       (* phi2 phi2)
       (-
        (*
         (* phi2 phi2)
         (- 0.008333333333333333 (* 0.0001984126984126984 (* phi2 phi2))))
        0.16666666666666666)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.55) {
		tmp = atan2((sin(lambda1) * 1.0), phi2);
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 + ((phi2 * phi2) * (((phi2 * phi2) * (0.008333333333333333 - (0.0001984126984126984 * (phi2 * phi2)))) - 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-1.55d0)) then
        tmp = atan2((sin(lambda1) * 1.0d0), phi2)
    else
        tmp = atan2((sin((lambda1 - lambda2)) * 1.0d0), (phi2 * (1.0d0 + ((phi2 * phi2) * (((phi2 * phi2) * (0.008333333333333333d0 - (0.0001984126984126984d0 * (phi2 * phi2)))) - 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.55) {
		tmp = Math.atan2((Math.sin(lambda1) * 1.0), phi2);
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 + ((phi2 * phi2) * (((phi2 * phi2) * (0.008333333333333333 - (0.0001984126984126984 * (phi2 * phi2)))) - 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -1.55:
		tmp = math.atan2((math.sin(lambda1) * 1.0), phi2)
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 + ((phi2 * phi2) * (((phi2 * phi2) * (0.008333333333333333 - (0.0001984126984126984 * (phi2 * phi2)))) - 0.16666666666666666)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.55)
		tmp = atan(Float64(sin(lambda1) * 1.0), phi2);
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * 1.0), Float64(phi2 * Float64(1.0 + Float64(Float64(phi2 * phi2) * Float64(Float64(Float64(phi2 * phi2) * Float64(0.008333333333333333 - Float64(0.0001984126984126984 * Float64(phi2 * phi2)))) - 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -1.55)
		tmp = atan2((sin(lambda1) * 1.0), phi2);
	else
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 + ((phi2 * phi2) * (((phi2 * phi2) * (0.008333333333333333 - (0.0001984126984126984 * (phi2 * phi2)))) - 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.55], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * 1.0), $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(N[(phi2 * phi2), $MachinePrecision] * N[(N[(N[(phi2 * phi2), $MachinePrecision] * N[(0.008333333333333333 - N[(0.0001984126984126984 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55000000000000004

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   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 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot 1}{\phi_2} \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot 1}{\phi_2} \]

   if -1.55000000000000004 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \color{blue}{{\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \color{blue}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)} \]

      14. pow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\phi_2 \cdot \phi_2\right)\right) - \frac{1}{6}\right)\right)} \]

      15. lift-*.f64 28.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - 0.16666666666666666\right)\right)} \]

   10. Applied rewrites 28.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \color{blue}{\left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 37: 31.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot 1}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.55)
   (atan2 (* (sin lambda1) 1.0) phi2)
   (atan2
    (* (sin (- lambda1 lambda2)) 1.0)
    (* phi2 (- 1.0 (* 0.16666666666666666 (* phi2 phi2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.55) {
		tmp = atan2((sin(lambda1) * 1.0), phi2);
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 - (0.16666666666666666 * (phi2 * 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) :: tmp
    if (phi2 <= (-1.55d0)) then
        tmp = atan2((sin(lambda1) * 1.0d0), phi2)
    else
        tmp = atan2((sin((lambda1 - lambda2)) * 1.0d0), (phi2 * (1.0d0 - (0.16666666666666666d0 * (phi2 * phi2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.55) {
		tmp = Math.atan2((Math.sin(lambda1) * 1.0), phi2);
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 - (0.16666666666666666 * (phi2 * phi2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -1.55:
		tmp = math.atan2((math.sin(lambda1) * 1.0), phi2)
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 - (0.16666666666666666 * (phi2 * phi2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.55)
		tmp = atan(Float64(sin(lambda1) * 1.0), phi2);
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * 1.0), Float64(phi2 * Float64(1.0 - Float64(0.16666666666666666 * Float64(phi2 * phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -1.55)
		tmp = atan2((sin(lambda1) * 1.0), phi2);
	else
		tmp = atan2((sin((lambda1 - lambda2)) * 1.0), (phi2 * (1.0 - (0.16666666666666666 * (phi2 * phi2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.55], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * 1.0), $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] / N[(phi2 * N[(1.0 - N[(0.16666666666666666 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55000000000000004

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   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 28.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

   11. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot 1}{\phi_2} \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\lambda_1} \cdot 1}{\phi_2} \]

   if -1.55000000000000004 < phi2

   1. Initial program 78.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. 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 48.6

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 48.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   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 31.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}\right)} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \color{blue}{{\phi_2}^{2}}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \color{blue}{{\phi_2}^{2}}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot {\phi_2}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot {\phi_2}^{\color{blue}{2}}\right)} \]

      6. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - \frac{1}{6} \cdot \left(\phi_2 \cdot \phi_2\right)\right)} \]

      7. lift-*.f64 28.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right)} \]

   10. Applied rewrites 28.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2 \cdot \color{blue}{\left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 38: 31.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), 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)), 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.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. 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 48.6

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 48.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

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 31.4%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

8. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
9. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

   2. lift--.f64 31.4

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

10. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
11. Add Preprocessing

Alternative 39: 28.6% 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 78.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. 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 48.6

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 48.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

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 31.4%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\sin \phi_2} \]

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 28.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 1}{\phi_2} \]

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 28.6

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \]

13. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025124 
(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))))))