Bearing on a great circle

Percentage Accurate: 79.5% → 99.7%

Time: 23.6s

Alternatives: 29

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \phi_2\\ \tan^{-1}_* \frac{\mathsf{fma}\left(-\cos \lambda_1, t\_0, \left(\sin \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \mathsf{fma}\left(\sin \lambda_1 \cdot \sin \phi_1, t\_0, \cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda2) (cos phi2))))
   (atan2
    (fma (- (cos lambda1)) t_0 (* (* (sin lambda1) (cos phi2)) (cos lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (fma
      (* (sin lambda1) (sin phi1))
      t_0
      (* (cos lambda2) (* (cos lambda1) (* (cos phi2) (sin phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda2) * cos(phi2);
	return atan2(fma(-cos(lambda1), t_0, ((sin(lambda1) * cos(phi2)) * cos(lambda2))), ((cos(phi1) * sin(phi2)) - fma((sin(lambda1) * sin(phi1)), t_0, (cos(lambda2) * (cos(lambda1) * (cos(phi2) * sin(phi1)))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda2) * cos(phi2))
	return atan(fma(Float64(-cos(lambda1)), t_0, Float64(Float64(sin(lambda1) * cos(phi2)) * cos(lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - fma(Float64(sin(lambda1) * sin(phi1)), t_0, Float64(cos(lambda2) * Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))))))
end

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 85.3

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

4. Applied rewrites 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-diff N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.8

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

   14. lift-*.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in lambda1 around inf

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

    1. associate-*r* N/A

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

    2. lower-fma.f64 N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-cos.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. lower-cos.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lower-sin.f64 N/A

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

    15. lower-cos.f64 N/A

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

    16. lower-cos.f64 99.8

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 85.3

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

4. Applied rewrites 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-diff N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in lambda1 around 0

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

   1. lower-atan2.f64 N/A

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

9. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 85.3

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

4. Applied rewrites 85.3%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-cos.f64 74.2

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

4. Applied rewrites 74.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. sin-diff N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

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

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

   11. lift-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.7

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 5: 89.3% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sin-diff N/A

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

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

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 85.3

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

4. Applied rewrites 85.3%

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

Alternative 6: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_2, -\cos \lambda_1, \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -4.4e-7) (not (<= phi1 6e-9)))
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (-
       t_0
       (*
        (* (sin phi1) (cos phi2))
        (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
     (atan2
      (*
       (fma (sin lambda2) (- (cos lambda1)) (* (cos lambda2) (sin lambda1)))
       (cos phi2))
      (- t_0 (* (cos (- lambda2 lambda1)) phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9)) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = atan2((fma(sin(lambda2), -cos(lambda1), (cos(lambda2) * sin(lambda1))) * cos(phi2)), (t_0 - (cos((lambda2 - lambda1)) * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9))
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = atan(Float64(fma(sin(lambda2), Float64(-cos(lambda1)), Float64(cos(lambda2) * sin(lambda1))) * cos(phi2)), Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * phi1)));
	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[Or[LessEqual[phi1, -4.4e-7], N[Not[LessEqual[phi1, 6e-9]], $MachinePrecision]], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision]) + N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.4000000000000002e-7 or 5.99999999999999996e-9 < phi1

   1. Initial program 69.4%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-cos.f64 70.1

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

   4. Applied rewrites 70.1%

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

   if -4.4000000000000002e-7 < phi1 < 5.99999999999999996e-9

   1. Initial program 78.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 78.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_2, -\cos \lambda_1, \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 85.3

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

4. Applied rewrites 85.3%

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

Alternative 8: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(t\_0 \cdot \sin \phi_1, -\cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_2, -\cos \lambda_1, \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (or (<= phi1 -4.4e-7) (not (<= phi1 6e-9)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (fma (* t_0 (sin phi1)) (- (cos phi2)) (* (sin phi2) (cos phi1))))
     (atan2
      (*
       (fma (sin lambda2) (- (cos lambda1)) (* (cos lambda2) (sin lambda1)))
       (cos phi2))
      (- (* (cos phi1) (sin phi2)) (* t_0 phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), fma((t_0 * sin(phi1)), -cos(phi2), (sin(phi2) * cos(phi1))));
	} else {
		tmp = atan2((fma(sin(lambda2), -cos(lambda1), (cos(lambda2) * sin(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (t_0 * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(Float64(t_0 * sin(phi1)), Float64(-cos(phi2)), Float64(sin(phi2) * cos(phi1))));
	else
		tmp = atan(Float64(fma(sin(lambda2), Float64(-cos(lambda1)), Float64(cos(lambda2) * sin(lambda1))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -4.4e-7], N[Not[LessEqual[phi1, 6e-9]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision]) + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $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[(t$95$0 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.4000000000000002e-7 or 5.99999999999999996e-9 < phi1

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.4

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

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

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 69.4%

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

   if -4.4000000000000002e-7 < phi1 < 5.99999999999999996e-9

   1. Initial program 78.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 78.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1, -\cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_2, -\cos \lambda_1, \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(t\_0 \cdot \sin \phi_1, -\cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (or (<= phi1 -4.4e-7) (not (<= phi1 6e-9)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (fma (* t_0 (sin phi1)) (- (cos phi2)) (* (sin phi2) (cos phi1))))
     (atan2
      (*
       (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
       (cos phi2))
      (- (* (cos phi1) (sin phi2)) (* t_0 phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), fma((t_0 * sin(phi1)), -cos(phi2), (sin(phi2) * cos(phi1))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (t_0 * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if ((phi1 <= -4.4e-7) || !(phi1 <= 6e-9))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(Float64(t_0 * sin(phi1)), Float64(-cos(phi2)), Float64(sin(phi2) * cos(phi1))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-cos(lambda1)) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -4.4e-7], N[Not[LessEqual[phi1, 6e-9]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision]) + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $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[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.4000000000000002e-7 or 5.99999999999999996e-9 < phi1

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.4

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

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

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 69.4%

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

   if -4.4000000000000002e-7 < phi1 < 5.99999999999999996e-9

   1. Initial program 78.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 78.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1, -\cos \phi_2, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.112 \lor \neg \left(\lambda_1 \leq 55000000000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (sin phi1) (cos phi2))))
   (if (or (<= lambda1 -0.112) (not (<= lambda1 55000000000000.0)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* t_1 (cos (- lambda1 lambda2)))))
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- t_0 (* t_1 (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double tmp;
	if ((lambda1 <= -0.112) || !(lambda1 <= 55000000000000.0)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(phi1) * cos(phi2)
    if ((lambda1 <= (-0.112d0)) .or. (.not. (lambda1 <= 55000000000000.0d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(phi1) * Math.cos(phi2);
	double tmp;
	if ((lambda1 <= -0.112) || !(lambda1 <= 55000000000000.0)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda2))));
	}
	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)
	tmp = 0
	if (lambda1 <= -0.112) or not (lambda1 <= 55000000000000.0):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (t_1 * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if ((lambda1 <= -0.112) || !(lambda1 <= 55000000000000.0))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))));
	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);
	tmp = 0.0;
	if ((lambda1 <= -0.112) || ~((lambda1 <= 55000000000000.0)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -0.112], N[Not[LessEqual[lambda1, 55000000000000.0]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.112000000000000002 or 5.5e13 < lambda1

   1. Initial program 49.4%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 51.0

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

   5. Applied rewrites 51.0%

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

   if -0.112000000000000002 < lambda1 < 5.5e13

   1. Initial program 96.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.112 \lor \neg \left(\lambda_1 \leq 55000000000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -5.8 \cdot 10^{+36} \lor \neg \left(\lambda_2 \leq 0.014\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (sin phi1) (cos phi2))))
   (if (or (<= lambda2 -5.8e+36) (not (<= lambda2 0.014)))
     (atan2 (* (sin (- lambda2)) (cos phi2)) (- t_0 (* t_1 (cos lambda2))))
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- t_0 (* t_1 (cos lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double tmp;
	if ((lambda2 <= -5.8e+36) || !(lambda2 <= 0.014)) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(phi1) * cos(phi2)
    if ((lambda2 <= (-5.8d+36)) .or. (.not. (lambda2 <= 0.014d0))) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))))
    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 tmp;
	if ((lambda2 <= -5.8e+36) || !(lambda2 <= 0.014)) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda2))));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda1))));
	}
	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)
	tmp = 0
	if (lambda2 <= -5.8e+36) or not (lambda2 <= 0.014):
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda2))))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if ((lambda2 <= -5.8e+36) || !(lambda2 <= 0.014))
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda1))));
	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);
	tmp = 0.0;
	if ((lambda2 <= -5.8e+36) || ~((lambda2 <= 0.014)))
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	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]}, If[Or[LessEqual[lambda2, -5.8e+36], N[Not[LessEqual[lambda2, 0.014]], $MachinePrecision]], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.8e36 or 0.0140000000000000003 < lambda2

   1. Initial program 56.2%

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

   3. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in lambda1 around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 54.5

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

   8. Applied rewrites 54.5%

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

   if -5.8e36 < lambda2 < 0.0140000000000000003

   1. Initial program 92.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5.8 \cdot 10^{+36} \lor \neg \left(\lambda_2 \leq 0.014\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.95 \cdot 10^{-20} \lor \neg \left(\lambda_1 \leq 2.4 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (sin phi1) (cos phi2))))
   (if (or (<= lambda1 -4.95e-20) (not (<= lambda1 2.4e-19)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* t_1 (cos (- lambda1 lambda2)))))
     (atan2 (* (sin (- lambda2)) (cos phi2)) (- t_0 (* t_1 (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double tmp;
	if ((lambda1 <= -4.95e-20) || !(lambda1 <= 2.4e-19)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(phi1) * cos(phi2)
    if ((lambda1 <= (-4.95d-20)) .or. (.not. (lambda1 <= 2.4d-19))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(phi1) * Math.cos(phi2);
	double tmp;
	if ((lambda1 <= -4.95e-20) || !(lambda1 <= 2.4e-19)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda2))));
	}
	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)
	tmp = 0
	if (lambda1 <= -4.95e-20) or not (lambda1 <= 2.4e-19):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (t_1 * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	tmp = 0.0
	if ((lambda1 <= -4.95e-20) || !(lambda1 <= 2.4e-19))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda2))));
	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);
	tmp = 0.0;
	if ((lambda1 <= -4.95e-20) || ~((lambda1 <= 2.4e-19)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - (t_1 * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -4.95e-20], N[Not[LessEqual[lambda1, 2.4e-19]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.95000000000000002e-20 or 2.40000000000000023e-19 < lambda1

   1. Initial program 52.7%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 53.0

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

   5. Applied rewrites 53.0%

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

   if -4.95000000000000002e-20 < lambda1 < 2.40000000000000023e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in lambda1 around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.95 \cdot 10^{-20} \lor \neg \left(\lambda_1 \leq 2.4 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.4 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 3.4e+14)
     (atan2
      (* (sin (- lambda1 lambda2)) (cos phi2))
      (- t_0 (* (sin phi1) (cos (- lambda1 lambda2)))))
     (atan2
      (* (sin (- lambda2)) (cos phi2))
      (- t_0 (* (* (sin phi1) (cos phi2)) (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= 3.4e+14) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if (lambda2 <= 3.4d+14) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= 3.4e+14) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= 3.4e+14:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= 3.4e+14)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= 3.4e+14)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((sin(-lambda2) * cos(phi2)), (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 3.4e+14], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.4e14

   1. Initial program 80.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 3.4e14 < lambda2

   1. Initial program 52.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in lambda1 around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 52.0

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

   8. Applied rewrites 52.0%

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

Alternative 14: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 73.9

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

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

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 73.9%

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

Alternative 15: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.245 \lor \neg \left(\phi_2 \leq 4700000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -0.245) (not (<= phi2 4700000000.0)))
     (atan2 (* t_1 (cos phi2)) (- t_0 (* (sin phi1) (cos lambda1))))
     (atan2
      (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
      (-
       t_0
       (*
        (fma (* phi2 phi2) -0.5 1.0)
        (* (cos (- lambda1 lambda2)) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.245) || !(phi2 <= 4700000000.0)) {
		tmp = atan2((t_1 * cos(phi2)), (t_0 - (sin(phi1) * cos(lambda1))));
	} else {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * t_1), (t_0 - (fma((phi2 * phi2), -0.5, 1.0) * (cos((lambda1 - lambda2)) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -0.245) || !(phi2 <= 4700000000.0))
		tmp = atan(Float64(t_1 * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(lambda1))));
	else
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * t_1), Float64(t_0 - Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -0.245], N[Not[LessEqual[phi2, 4700000000.0]], $MachinePrecision]], N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.245 or 4.7e9 < phi2

   1. Initial program 68.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 45.6%

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

   if -0.245 < phi2 < 4.7e9

   1. Initial program 79.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in phi2 around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

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

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

       10. lower-sin.f64 N/A

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

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

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

       12. metadata-eval N/A

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

       13. *-lft-identity N/A

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

       14. lower--.f64 51.4

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

   11. Applied rewrites 51.4%

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

   12. Taylor expanded in phi2 around 0

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

       1. associate-*r* N/A

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

       2. distribute-lft1-in N/A

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

       3. +-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-lft-identity N/A

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

       12. metadata-eval N/A

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

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

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

       14. lower-cos.f64 N/A

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

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

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

       16. metadata-eval N/A

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

       17. *-lft-identity N/A

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

       18. lower--.f64 N/A

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

       19. lower-sin.f64 78.7

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

   14. Applied rewrites 78.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.245 \lor \neg \left(\phi_2 \leq 4700000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.7% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. *-lft-identity N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-cos.f64 N/A

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

   8. metadata-eval N/A

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

   9. cancel-sign-sub-inv N/A

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

   10. *-lft-identity N/A

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

   11. lower--.f64 61.3

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

5. Applied rewrites 61.3%

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

Alternative 17: 61.8% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. *-lft-identity N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-cos.f64 N/A

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

   8. metadata-eval N/A

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

   9. cancel-sign-sub-inv N/A

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

   10. *-lft-identity N/A

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

   11. lower--.f64 61.3

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

5. Applied rewrites 61.3%

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

6. Taylor expanded in lambda1 around 0

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

8. Applied rewrites 59.3%

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

Alternative 18: 63.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.84:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3250000000:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \sin \left(\lambda_2 - \left(\lambda_1 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi2 -0.84)
     (atan2
      t_2
      (-
       t_0
       (*
        (*
         (fma -0.16666666666666666 (* phi1 phi1) 1.0)
         (cos (- lambda2 lambda1)))
        phi1)))
     (if (<= phi2 3250000000.0)
       (atan2
        (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
        (-
         t_0
         (*
          (fma (* phi2 phi2) -0.5 1.0)
          (* (cos (- lambda1 lambda2)) (sin phi1)))))
       (atan2
        t_2
        (- t_0 (* (sin (- lambda2 (- lambda1 (/ (PI) 2.0)))) phi1)))))))

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.839999999999999969

   1. Initial program 66.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \color{blue}{\phi_1}} \]

   if -0.839999999999999969 < phi2 < 3.25e9

   1. Initial program 79.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in phi2 around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

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

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

       10. lower-sin.f64 N/A

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

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

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

       12. metadata-eval N/A

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

       13. *-lft-identity N/A

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

       14. lower--.f64 51.4

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

   11. Applied rewrites 51.4%

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

   12. Taylor expanded in phi2 around 0

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

       1. associate-*r* N/A

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

       2. distribute-lft1-in N/A

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

       3. +-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-lft-identity N/A

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

       12. metadata-eval N/A

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

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

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

       14. lower-cos.f64 N/A

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

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

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

       16. metadata-eval N/A

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

       17. *-lft-identity N/A

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

       18. lower--.f64 N/A

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

       19. lower-sin.f64 78.7

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

   14. Applied rewrites 78.7%

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

   if 3.25e9 < phi2

   1. Initial program 70.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 43.4%

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

   10. Applied rewrites 43.4%

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

Alternative 19: 64.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.8 \lor \neg \left(\phi_2 \leq 3250000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \sin \left(\lambda_2 - \left(\lambda_1 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -0.8) (not (<= phi2 3250000000.0)))
     (atan2
      (* t_1 (cos phi2))
      (- t_0 (* (sin (- lambda2 (- lambda1 (/ (PI) 2.0)))) phi1)))
     (atan2
      (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
      (- t_0 (* (sin phi1) (cos (- lambda1 lambda2))))))))

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.80000000000000004 or 3.25e9 < phi2

   1. Initial program 68.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 41.1%

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

   if -0.80000000000000004 < phi2 < 3.25e9

   1. Initial program 79.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower--.f64 78.5

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

   8. Applied rewrites 78.5%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.8 \lor \neg \left(\phi_2 \leq 3250000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \left(\lambda_2 - \left(\lambda_1 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.58:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3250000000:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \sin \left(\lambda_2 - \left(\lambda_1 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi2 -0.58)
     (atan2
      t_2
      (-
       t_0
       (*
        (*
         (fma -0.16666666666666666 (* phi1 phi1) 1.0)
         (cos (- lambda2 lambda1)))
        phi1)))
     (if (<= phi2 3250000000.0)
       (atan2
        (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
        (- t_0 (* (sin phi1) (cos (- lambda1 lambda2)))))
       (atan2
        t_2
        (- t_0 (* (sin (- lambda2 (- lambda1 (/ (PI) 2.0)))) phi1)))))))

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.57999999999999996

   1. Initial program 66.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \color{blue}{\phi_1}} \]

   if -0.57999999999999996 < phi2 < 3.25e9

   1. Initial program 79.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower--.f64 78.5

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

   8. Applied rewrites 78.5%

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

   if 3.25e9 < phi2

   1. Initial program 70.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 43.4%

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

   10. Applied rewrites 43.4%

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

Alternative 21: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.8:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 0.06:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi2 -0.8)
     (atan2 t_2 (- t_0 (* (cos lambda1) phi1)))
     (if (<= phi2 0.06)
       (atan2
        (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
        (- t_0 (* (sin phi1) (cos (- lambda1 lambda2)))))
       (atan2 t_2 (- t_0 (* (cos (- lambda2 lambda1)) phi1)))))))

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 = t_1 * cos(phi2);
	double tmp;
	if (phi2 <= -0.8) {
		tmp = atan2(t_2, (t_0 - (cos(lambda1) * phi1)));
	} else if (phi2 <= 0.06) {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * t_1), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2(t_2, (t_0 - (cos((lambda2 - lambda1)) * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -0.8)
		tmp = atan(t_2, Float64(t_0 - Float64(cos(lambda1) * phi1)));
	elseif (phi2 <= 0.06)
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * t_1), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.8], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.06], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.80000000000000004

   1. Initial program 66.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 43.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \phi_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \]

   if -0.80000000000000004 < phi2 < 0.059999999999999998

   1. Initial program 80.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 79.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower--.f64 79.5

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Applied rewrites 79.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 0.059999999999999998 < phi2

   1. Initial program 70.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 48.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.8:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 0.058:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi2 -0.8)
     (atan2 t_2 (- t_0 (* (cos lambda1) phi1)))
     (if (<= phi2 0.058)
       (atan2
        (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
        (- t_0 (* (cos lambda2) (sin phi1))))
       (atan2 t_2 (- t_0 (* (cos (- lambda2 lambda1)) phi1)))))))

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 = t_1 * cos(phi2);
	double tmp;
	if (phi2 <= -0.8) {
		tmp = atan2(t_2, (t_0 - (cos(lambda1) * phi1)));
	} else if (phi2 <= 0.058) {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * t_1), (t_0 - (cos(lambda2) * sin(phi1))));
	} else {
		tmp = atan2(t_2, (t_0 - (cos((lambda2 - lambda1)) * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -0.8)
		tmp = atan(t_2, Float64(t_0 - Float64(cos(lambda1) * phi1)));
	elseif (phi2 <= 0.058)
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * t_1), Float64(t_0 - Float64(cos(lambda2) * sin(phi1))));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.8], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.058], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.80000000000000004

   1. Initial program 66.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 43.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \phi_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \]

   if -0.80000000000000004 < phi2 < 0.0580000000000000029

   1. Initial program 80.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 79.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       2. distribute-rgt1-in N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       5. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       7. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       8. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       9. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       10. lower-sin.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       11. fp-cancel-sign-sub-inv N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       12. metadata-eval N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       13. *-lft-identity N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       14. lower--.f64 51.9

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   11. Applied rewrites 51.9%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\sin \phi_1}} \]
   13. Step-by-step derivation

   14. Applied rewrites 74.8%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \color{blue}{\sin \phi_1}} \]

   if 0.0580000000000000029 < phi2

   1. Initial program 70.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 48.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.8:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 0.06:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* t_1 (cos phi2))))
   (if (<= phi2 -0.8)
     (atan2 t_2 (- t_0 (* (cos lambda1) phi1)))
     (if (<= phi2 0.06)
       (atan2
        (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
        (- t_0 (* (cos lambda1) (sin phi1))))
       (atan2 t_2 (- t_0 (* (cos (- lambda2 lambda1)) phi1)))))))

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 = t_1 * cos(phi2);
	double tmp;
	if (phi2 <= -0.8) {
		tmp = atan2(t_2, (t_0 - (cos(lambda1) * phi1)));
	} else if (phi2 <= 0.06) {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * t_1), (t_0 - (cos(lambda1) * sin(phi1))));
	} else {
		tmp = atan2(t_2, (t_0 - (cos((lambda2 - lambda1)) * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(t_1 * cos(phi2))
	tmp = 0.0
	if (phi2 <= -0.8)
		tmp = atan(t_2, Float64(t_0 - Float64(cos(lambda1) * phi1)));
	elseif (phi2 <= 0.06)
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * t_1), Float64(t_0 - Float64(cos(lambda1) * sin(phi1))));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.8], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.06], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.80000000000000004

   1. Initial program 66.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 43.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 43.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \phi_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \]

   if -0.80000000000000004 < phi2 < 0.059999999999999998

   1. Initial program 80.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 79.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       2. distribute-rgt1-in N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       5. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       7. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       8. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       9. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       10. lower-sin.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       11. fp-cancel-sign-sub-inv N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       12. metadata-eval N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       13. *-lft-identity N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       14. lower--.f64 51.9

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   11. Applied rewrites 51.9%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\sin \phi_1}} \]
   13. Step-by-step derivation

   14. Applied rewrites 67.9%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \color{blue}{\sin \phi_1}} \]

   if 0.059999999999999998 < phi2

   1. Initial program 70.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 48.4

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 50.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.95 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= phi1 -1.95e-31)
     (atan2
      (* (fma -0.5 (* phi2 phi2) 1.0) t_1)
      (-
       t_0
       (*
        (*
         (fma -0.16666666666666666 (* phi1 phi1) 1.0)
         (cos (- lambda1 lambda2)))
        phi1)))
     (atan2 (* t_1 (cos phi2)) (- t_0 (* (cos lambda1) phi1))))))

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 tmp;
	if (phi1 <= -1.95e-31) {
		tmp = atan2((fma(-0.5, (phi2 * phi2), 1.0) * t_1), (t_0 - ((fma(-0.16666666666666666, (phi1 * phi1), 1.0) * cos((lambda1 - lambda2))) * phi1)));
	} else {
		tmp = atan2((t_1 * cos(phi2)), (t_0 - (cos(lambda1) * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -1.95e-31)
		tmp = atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * t_1), Float64(t_0 - Float64(Float64(fma(-0.16666666666666666, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda1 - lambda2))) * phi1)));
	else
		tmp = atan(Float64(t_1 * cos(phi2)), Float64(t_0 - Float64(cos(lambda1) * phi1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.95e-31], N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[(N[(-0.16666666666666666 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.9500000000000001e-31

   1. Initial program 67.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 44.6

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 44.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       2. distribute-rgt1-in N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       5. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       7. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       8. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       9. fp-cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       10. lower-sin.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       11. fp-cancel-sign-sub-inv N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       12. metadata-eval N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       13. *-lft-identity N/A

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

       14. lower--.f64 10.6

           \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   11. Applied rewrites 10.6%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   13. Step-by-step derivation

   14. Applied rewrites 15.6%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\phi_1}} \]

   if -1.9500000000000001e-31 < phi1

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

      10. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      11. lower--.f64 67.8

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   5. Applied rewrites 67.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \phi_1} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 49.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (- (* (cos phi1) (sin phi2)) (* (cos (- lambda2 lambda1)) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda2 - lambda1)) * phi1)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda2 - lambda1)) * phi1)))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos((lambda2 - lambda1)) * phi1)));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (math.cos((lambda2 - lambda1)) * phi1)))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(Float64(lambda2 - lambda1)) * phi1)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda2 - lambda1)) * phi1)));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

   7. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

   10. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   11. lower--.f64 61.3

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

5. Applied rewrites 61.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 45.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]
9. Add Preprocessing

Alternative 26: 31.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (fma -0.5 (* phi2 phi2) 1.0) (sin (- lambda1 lambda2)))
  (- (* (cos phi1) (sin phi2)) (* (cos (- lambda2 lambda1)) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(-0.5, (phi2 * phi2), 1.0) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos((lambda2 - lambda1)) * phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(Float64(lambda2 - lambda1)) * phi1)))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

   7. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

   10. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   11. lower--.f64 61.3

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

5. Applied rewrites 61.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 45.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    4. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    5. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    7. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    8. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    9. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    10. lower-sin.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    11. fp-cancel-sign-sub-inv N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    13. *-lft-identity N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    14. lower--.f64 28.6

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

11. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
12. Add Preprocessing

Alternative 27: 30.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (fma -0.5 (* phi2 phi2) 1.0) (sin (- lambda1 lambda2)))
  (- (* (cos phi1) (sin phi2)) (* (cos lambda2) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(-0.5, (phi2 * phi2), 1.0) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda2) * phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda2) * phi1)))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

   7. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

   10. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   11. lower--.f64 61.3

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

5. Applied rewrites 61.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 45.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    4. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    5. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    7. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    8. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    9. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    10. lower-sin.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    11. fp-cancel-sign-sub-inv N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    13. *-lft-identity N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    14. lower--.f64 28.6

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

11. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

12. Taylor expanded in lambda1 around 0

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \phi_1} \]
13. Step-by-step derivation

14. Applied rewrites 28.4%

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \phi_1} \]
15. Add Preprocessing

Alternative 28: 30.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (fma -0.5 (* phi2 phi2) 1.0) (sin (- lambda1 lambda2)))
  (- (* (cos phi1) (sin phi2)) (* (cos lambda1) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(-0.5, (phi2 * phi2), 1.0) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(-0.5, Float64(phi2 * phi2), 1.0) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * phi1)))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

   7. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

   10. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   11. lower--.f64 61.3

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

5. Applied rewrites 61.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 45.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    4. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    5. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    7. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    8. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    9. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    10. lower-sin.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    11. fp-cancel-sign-sub-inv N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    13. *-lft-identity N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    14. lower--.f64 28.6

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

11. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \phi_1} \]
13. Step-by-step derivation

14. Applied rewrites 28.2%

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1} \]
15. Add Preprocessing

Alternative 29: 22.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (fma (* phi2 phi2) -0.5 1.0) (sin lambda1))
  (- (* (cos phi1) (sin phi2)) (* (cos (- lambda2 lambda1)) phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma((phi2 * phi2), -0.5, 1.0) * sin(lambda1)), ((cos(phi1) * sin(phi2)) - (cos((lambda2 - lambda1)) * phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * sin(lambda1)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(Float64(lambda2 - lambda1)) * phi1)))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(1\right)\right) \cdot \lambda_2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{-1} \cdot \lambda_2\right)} \]

   7. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \lambda_2\right)} \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - 1 \cdot \lambda_2\right)}} \]

   10. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   11. lower--.f64 61.3

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

5. Applied rewrites 61.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 45.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\phi_1}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    4. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    5. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    7. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    8. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    9. fp-cancel-sign-sub-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    10. lower-sin.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \color{blue}{\sin \left(\lambda_1 + -1 \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    11. fp-cancel-sign-sub-inv N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{1} \cdot \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    13. *-lft-identity N/A

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \color{blue}{\lambda_2}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

    14. lower--.f64 28.6

        \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

11. Applied rewrites 28.6%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
13. Step-by-step derivation

14. Applied rewrites 21.1%

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \color{blue}{\sin \lambda_1}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \phi_1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(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))))))