Midpoint on a great circle

Percentage Accurate: 98.6% → 99.6%

Time: 15.0s

Alternatives: 22

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(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 = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(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 = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\ t_2 := \cos \lambda_2 \cdot \cos \lambda_1\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{t\_0}\right) \cdot t\_0\right)}{\cos \phi_1 + \cos \phi_2 \cdot \frac{{t\_1}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2 - t\_1 \cdot t\_2\right)}} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2)))
        (t_1 (* (sin lambda1) (sin lambda2)))
        (t_2 (* (cos lambda2) (cos lambda1))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0))
     (+
      (cos phi1)
      (*
       (cos phi2)
       (/
        (+ (pow t_1 3.0) (pow t_2 3.0))
        (fma t_1 t_1 (- (* t_2 t_2) (* t_1 t_2))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = sin(lambda1) * sin(lambda2);
	double t_2 = cos(lambda2) * cos(lambda1);
	return lambda1 + atan2((cos(phi2) * ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(phi1) + (cos(phi2) * ((pow(t_1, 3.0) + pow(t_2, 3.0)) / fma(t_1, t_1, ((t_2 * t_2) - (t_1 * t_2)))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = Float64(sin(lambda1) * sin(lambda2))
	t_2 = Float64(cos(lambda2) * cos(lambda1))
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), Float64(cos(phi1) + Float64(cos(phi2) * Float64(Float64((t_1 ^ 3.0) + (t_2 ^ 3.0)) / fma(t_1, t_1, Float64(Float64(t_2 * t_2) - Float64(t_1 * t_2))))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. flip3-+ N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-+.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-unsound-pow.f64 N/A

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

   9. lower-unsound-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \frac{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3} + {\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\sin \lambda_2 \cdot \sin \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) - \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}}} \]

7. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0))
     (+
      (cos phi1)
      (*
       (cos phi2)
       (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	return lambda1 + atan2((cos(phi2) * ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(phi1) + (cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), Float64(cos(phi1) + Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1)))))))
end

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-mult-rev N/A

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

   5. lower--.f64 99.6

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 4: 98.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0))
     (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	return lambda1 + atan2((cos(phi2) * ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(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
    real(8) :: t_0
    t_0 = sin(lambda1) * cos(lambda2)
    code = lambda1 + atan2((cos(phi2) * ((1.0d0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(lambda1) * Math.cos(lambda2);
	return lambda1 + Math.atan2((Math.cos(phi2) * ((1.0 - ((Math.sin(lambda2) * Math.cos(lambda1)) / t_0)) * t_0)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(lambda1) * math.cos(lambda2)
	return lambda1 + math.atan2((math.cos(phi2) * ((1.0 - ((math.sin(lambda2) * math.cos(lambda1)) / t_0)) * t_0)), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(lambda1) * cos(lambda2);
	tmp = lambda1 + atan2((cos(phi2) * ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

Alternative 5: 98.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (*
     (- 1.0 (* (/ (cos lambda1) (sin lambda1)) (tan lambda2)))
     (* (sin lambda1) (cos lambda2))))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * ((1.0 - ((cos(lambda1) / sin(lambda1)) * tan(lambda2))) * (sin(lambda1) * cos(lambda2)))), (cos(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 = lambda1 + atan2((cos(phi2) * ((1.0d0 - ((cos(lambda1) / sin(lambda1)) * tan(lambda2))) * (sin(lambda1) * cos(lambda2)))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * ((1.0 - ((Math.cos(lambda1) / Math.sin(lambda1)) * Math.tan(lambda2))) * (Math.sin(lambda1) * Math.cos(lambda2)))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * ((1.0 - ((math.cos(lambda1) / math.sin(lambda1)) * math.tan(lambda2))) * (math.sin(lambda1) * math.cos(lambda2)))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(1.0 - Float64(Float64(cos(lambda1) / sin(lambda1)) * tan(lambda2))) * Float64(sin(lambda1) * cos(lambda2)))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * ((1.0 - ((cos(lambda1) / sin(lambda1)) * tan(lambda2))) * (sin(lambda1) * cos(lambda2)))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Cos[lambda1], $MachinePrecision] / N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Tan[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. quot-tan N/A

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

   11. lower-tan.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 6: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * ((1.0 - (sin(lambda2) / (lambda1 * cos(lambda2)))) * (sin(lambda1) * cos(lambda2)))), (cos(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 = lambda1 + atan2((cos(phi2) * ((1.0d0 - (sin(lambda2) / (lambda1 * cos(lambda2)))) * (sin(lambda1) * cos(lambda2)))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * ((1.0 - (Math.sin(lambda2) / (lambda1 * Math.cos(lambda2)))) * (Math.sin(lambda1) * Math.cos(lambda2)))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * ((1.0 - (math.sin(lambda2) / (lambda1 * math.cos(lambda2)))) * (math.sin(lambda1) * math.cos(lambda2)))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(1.0 - Float64(sin(lambda2) / Float64(lambda1 * cos(lambda2)))) * Float64(sin(lambda1) * cos(lambda2)))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * ((1.0 - (sin(lambda2) / (lambda1 * cos(lambda2)))) * (sin(lambda1) * cos(lambda2)))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(1.0 - N[(N[Sin[lambda2], $MachinePrecision] / N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in lambda1 around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 98.5

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

6. Applied rewrites 98.5%

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

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 98.6

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

3. Applied rewrites 98.6%

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

Alternative 8: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t\_0}{t\_1}\\ t_3 := \lambda_1 + t\_2\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{if}\;t\_3 \leq -3.14:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_1 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))
        (t_2 (atan2 t_0 t_1))
        (t_3 (+ lambda1 t_2))
        (t_4
         (+ lambda1 (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos lambda1)))))))
   (if (<= t_3 -3.14)
     (+ lambda1 (atan2 (* (cos phi2) (sin lambda1)) t_1))
     (if (<= t_3 -0.02)
       t_2
       (if (<= t_3 4e-8) t_4 (if (<= t_3 3.0) t_2 t_4))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double t_1 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	double t_2 = atan2(t_0, t_1);
	double t_3 = lambda1 + t_2;
	double t_4 = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	double tmp;
	if (t_3 <= -3.14) {
		tmp = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_1);
	} else if (t_3 <= -0.02) {
		tmp = t_2;
	} else if (t_3 <= 4e-8) {
		tmp = t_4;
	} else if (t_3 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    t_1 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))
    t_2 = atan2(t_0, t_1)
    t_3 = lambda1 + t_2
    t_4 = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))))
    if (t_3 <= (-3.14d0)) then
        tmp = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_1)
    else if (t_3 <= (-0.02d0)) then
        tmp = t_2
    else if (t_3 <= 4d-8) then
        tmp = t_4
    else if (t_3 <= 3.0d0) then
        tmp = t_2
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)));
	double t_2 = Math.atan2(t_0, t_1);
	double t_3 = lambda1 + t_2;
	double t_4 = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda1))));
	double tmp;
	if (t_3 <= -3.14) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(lambda1)), t_1);
	} else if (t_3 <= -0.02) {
		tmp = t_2;
	} else if (t_3 <= 4e-8) {
		tmp = t_4;
	} else if (t_3 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))
	t_2 = math.atan2(t_0, t_1)
	t_3 = lambda1 + t_2
	t_4 = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda1))))
	tmp = 0
	if t_3 <= -3.14:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(lambda1)), t_1)
	elif t_3 <= -0.02:
		tmp = t_2
	elif t_3 <= 4e-8:
		tmp = t_4
	elif t_3 <= 3.0:
		tmp = t_2
	else:
		tmp = t_4
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	t_2 = atan(t_0, t_1)
	t_3 = Float64(lambda1 + t_2)
	t_4 = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda1)))))
	tmp = 0.0
	if (t_3 <= -3.14)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(lambda1)), t_1));
	elseif (t_3 <= -0.02)
		tmp = t_2;
	elseif (t_3 <= 4e-8)
		tmp = t_4;
	elseif (t_3 <= 3.0)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	t_1 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	t_2 = atan2(t_0, t_1);
	t_3 = lambda1 + t_2;
	t_4 = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	tmp = 0.0;
	if (t_3 <= -3.14)
		tmp = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_1);
	elseif (t_3 <= -0.02)
		tmp = t_2;
	elseif (t_3 <= 4e-8)
		tmp = t_4;
	elseif (t_3 <= 3.0)
		tmp = t_2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$0 / t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -3.14], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.02], t$95$2, If[LessEqual[t$95$3, 4e-8], t$95$4, If[LessEqual[t$95$3, 3.0], t$95$2, t$95$4]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -3.14000000000000012

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   if -3.14000000000000012 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -0.0200000000000000004 or 4.0000000000000001e-8 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 3

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower--.f64 41.7

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

   4. Applied rewrites 41.7%

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

   if -0.0200000000000000004 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 4.0000000000000001e-8 or 3 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

Alternative 9: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * 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 = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos(lambda2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate-+l+ N/A

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

   6. sin-sum N/A

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

   7. sin-+PI/2-rev N/A

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

   8. cos-neg-rev N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. mult-flip N/A

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

   15. lower-*.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cos.f64 N/A

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

   21. lower-cos.f64 98.6

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in lambda1 around 0

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

   1. lower-cos.f64 97.8

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

6. Applied rewrites 97.8%

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

Alternative 10: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t\_0}\\ \mathbf{if}\;\lambda_1 \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))
        (t_1 (+ lambda1 (atan2 (* (cos phi2) (sin lambda1)) t_0))))
   (if (<= lambda1 -2.9e-8)
     t_1
     (if (<= lambda1 9.5e-18)
       (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) t_0)
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	double t_1 = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_0);
	double tmp;
	if (lambda1 <= -2.9e-8) {
		tmp = t_1;
	} else if (lambda1 <= 9.5e-18) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))
    t_1 = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_0)
    if (lambda1 <= (-2.9d-8)) then
        tmp = t_1
    else if (lambda1 <= 9.5d-18) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)));
	double t_1 = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(lambda1)), t_0);
	double tmp;
	if (lambda1 <= -2.9e-8) {
		tmp = t_1;
	} else if (lambda1 <= 9.5e-18) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))
	t_1 = lambda1 + math.atan2((math.cos(phi2) * math.sin(lambda1)), t_0)
	tmp = 0
	if lambda1 <= -2.9e-8:
		tmp = t_1
	elif lambda1 <= 9.5e-18:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	t_1 = Float64(lambda1 + atan(Float64(cos(phi2) * sin(lambda1)), t_0))
	tmp = 0.0
	if (lambda1 <= -2.9e-8)
		tmp = t_1;
	elseif (lambda1 <= 9.5e-18)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	t_1 = lambda1 + atan2((cos(phi2) * sin(lambda1)), t_0);
	tmp = 0.0;
	if (lambda1 <= -2.9e-8)
		tmp = t_1;
	elseif (lambda1 <= 9.5e-18)
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -2.9e-8], t$95$1, If[LessEqual[lambda1, 9.5e-18], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.9000000000000002e-8 or 9.5000000000000003e-18 < lambda1

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   if -2.9000000000000002e-8 < lambda1 < 9.5000000000000003e-18

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower--.f64 41.7

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

   4. Applied rewrites 41.7%

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

Alternative 11: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 + t\_1\\ t_3 := \cos \left(\left|\phi_2\right|\right)\\ t_4 := t\_3 \cdot t\_0\\ \mathbf{if}\;\lambda_1 \leq -0.2:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \left|\phi_2\right|\right)\right) \cdot t\_0}{t\_2}\\ \mathbf{elif}\;\lambda_1 \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_4}{\cos \phi_1 + t\_3 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_4}{t\_2}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (+ (cos phi1) t_1))
        (t_3 (cos (fabs phi2)))
        (t_4 (* t_3 t_0)))
   (if (<= lambda1 -0.2)
     (+ lambda1 (atan2 (* (sin (fma 0.5 PI (fabs phi2))) t_0) t_2))
     (if (<= lambda1 3.5e-13)
       (atan2 t_4 (+ (cos phi1) (* t_3 t_1)))
       (+ lambda1 (atan2 t_4 t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi1) + t_1;
	double t_3 = cos(fabs(phi2));
	double t_4 = t_3 * t_0;
	double tmp;
	if (lambda1 <= -0.2) {
		tmp = lambda1 + atan2((sin(fma(0.5, ((double) M_PI), fabs(phi2))) * t_0), t_2);
	} else if (lambda1 <= 3.5e-13) {
		tmp = atan2(t_4, (cos(phi1) + (t_3 * t_1)));
	} else {
		tmp = lambda1 + atan2(t_4, t_2);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi1) + t_1)
	t_3 = cos(abs(phi2))
	t_4 = Float64(t_3 * t_0)
	tmp = 0.0
	if (lambda1 <= -0.2)
		tmp = Float64(lambda1 + atan(Float64(sin(fma(0.5, pi, abs(phi2))) * t_0), t_2));
	elseif (lambda1 <= 3.5e-13)
		tmp = atan(t_4, Float64(cos(phi1) + Float64(t_3 * t_1)));
	else
		tmp = Float64(lambda1 + atan(t_4, t_2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0), $MachinePrecision]}, If[LessEqual[lambda1, -0.2], N[(lambda1 + N[ArcTan[N[(N[Sin[N[(0.5 * Pi + N[Abs[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.5e-13], N[ArcTan[t$95$4 / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$4 / t$95$2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.20000000000000001

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 75.6

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

   6. Applied rewrites 75.6%

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

   if -0.20000000000000001 < lambda1 < 3.5000000000000002e-13

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower--.f64 41.7

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

   4. Applied rewrites 41.7%

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

   if 3.5000000000000002e-13 < lambda1

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 12: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\left|\phi_2\right|\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 7500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\left(-\left|\phi_2\right|\right) + 0.5 \cdot \pi\right) \cdot t\_0}{\cos \phi_1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{1 + \mathsf{fma}\left(-0.5, {\phi_1}^{2}, t\_2 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (cos (fabs phi2))))
   (if (<= (fabs phi2) 7500000.0)
     (+
      lambda1
      (atan2 (* (sin (+ (- (fabs phi2)) (* 0.5 PI))) t_0) (+ (cos phi1) t_1)))
     (+
      lambda1
      (atan2 (* t_2 t_0) (+ 1.0 (fma -0.5 (pow phi1 2.0) (* t_2 t_1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(fabs(phi2));
	double tmp;
	if (fabs(phi2) <= 7500000.0) {
		tmp = lambda1 + atan2((sin((-fabs(phi2) + (0.5 * ((double) M_PI)))) * t_0), (cos(phi1) + t_1));
	} else {
		tmp = lambda1 + atan2((t_2 * t_0), (1.0 + fma(-0.5, pow(phi1, 2.0), (t_2 * t_1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = cos(abs(phi2))
	tmp = 0.0
	if (abs(phi2) <= 7500000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(Float64(Float64(-abs(phi2)) + Float64(0.5 * pi))) * t_0), Float64(cos(phi1) + t_1)));
	else
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(1.0 + fma(-0.5, (phi1 ^ 2.0), Float64(t_2 * t_1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 7500000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[N[((-N[Abs[phi2], $MachinePrecision]) + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.5e6

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 75.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.7

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

   6. Applied rewrites 75.7%

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

   if 7.5e6 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 79.4

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \mathsf{fma}\left(-0.5, {\phi_1}^{2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   4. Applied rewrites 79.4%

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

Alternative 13: 88.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\left|\phi_1\right| \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 + \cos \phi_2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \left(\left|\phi_1\right|\right) + t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (fabs phi1) 4.6e-7)
     (+ lambda1 (atan2 t_1 (+ 1.0 (* (cos phi2) t_0))))
     (+ lambda1 (atan2 t_1 (+ (cos (fabs phi1)) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (fabs(phi1) <= 4.6e-7) {
		tmp = lambda1 + atan2(t_1, (1.0 + (cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(fabs(phi1)) + t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (abs(phi1) <= 4.6d-7) then
        tmp = lambda1 + atan2(t_1, (1.0d0 + (cos(phi2) * t_0)))
    else
        tmp = lambda1 + atan2(t_1, (cos(abs(phi1)) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.abs(phi1) <= 4.6e-7) {
		tmp = lambda1 + Math.atan2(t_1, (1.0 + (Math.cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(Math.abs(phi1)) + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.fabs(phi1) <= 4.6e-7:
		tmp = lambda1 + math.atan2(t_1, (1.0 + (math.cos(phi2) * t_0)))
	else:
		tmp = lambda1 + math.atan2(t_1, (math.cos(math.fabs(phi1)) + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (abs(phi1) <= 4.6e-7)
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 + Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(abs(phi1)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (abs(phi1) <= 4.6e-7)
		tmp = lambda1 + atan2(t_1, (1.0 + (cos(phi2) * t_0)));
	else
		tmp = lambda1 + atan2(t_1, (cos(abs(phi1)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[phi1], $MachinePrecision], 4.6e-7], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[N[Abs[phi1], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 4.5999999999999999e-7

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 78.0%

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

   if 4.5999999999999999e-7 < phi1

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 14: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.65:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.65)
     (+ lambda1 (atan2 t_1 (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) t_0)))
     (+ lambda1 (atan2 t_1 (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.65) {
		tmp = lambda1 + atan2(t_1, ((1.0 + (-0.5 * pow(phi1, 2.0))) + t_0));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.65d0) then
        tmp = lambda1 + atan2(t_1, ((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) + t_0))
    else
        tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.65) {
		tmp = lambda1 + Math.atan2(t_1, ((1.0 + (-0.5 * Math.pow(phi1, 2.0))) + t_0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.65:
		tmp = lambda1 + math.atan2(t_1, ((1.0 + (-0.5 * math.pow(phi1, 2.0))) + t_0))
	else:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.65)
		tmp = Float64(lambda1 + atan(t_1, Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + t_0)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.65)
		tmp = lambda1 + atan2(t_1, ((1.0 + (-0.5 * (phi1 ^ 2.0))) + t_0));
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.65], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.650000000000000022

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 68.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   if 0.650000000000000022 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 15: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.65:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \left(-\lambda_2\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.65)
     (+
      lambda1
      (atan2
       t_0
       (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) (cos (- lambda1 lambda2)))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.65) {
		tmp = lambda1 + atan2(t_0, ((1.0 + (-0.5 * pow(phi1, 2.0))) + cos((lambda1 - lambda2))));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(-lambda2)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.65d0) then
        tmp = lambda1 + atan2(t_0, ((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) + cos((lambda1 - lambda2))))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(-lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.65) {
		tmp = lambda1 + Math.atan2(t_0, ((1.0 + (-0.5 * Math.pow(phi1, 2.0))) + Math.cos((lambda1 - lambda2))));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos(-lambda2)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.65:
		tmp = lambda1 + math.atan2(t_0, ((1.0 + (-0.5 * math.pow(phi1, 2.0))) + math.cos((lambda1 - lambda2))))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos(-lambda2)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.65)
		tmp = Float64(lambda1 + atan(t_0, Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + cos(Float64(lambda1 - lambda2)))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(-lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.65)
		tmp = lambda1 + atan2(t_0, ((1.0 + (-0.5 * (phi1 ^ 2.0))) + cos((lambda1 - lambda2))));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(-lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.65], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.650000000000000022

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 68.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

   if 0.650000000000000022 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in lambda1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 77.3

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

   10. Applied rewrites 77.3%

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

Alternative 16: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 31000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\left(\left|\phi_2\right|\right)}^{2}\right) \cdot t\_0}{\cos \phi_1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \left(\left|\phi_2\right|\right) \cdot t\_0}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + t\_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= (fabs phi2) 31000000000.0)
     (+
      lambda1
      (atan2
       (* (+ 1.0 (* -0.5 (pow (fabs phi2) 2.0))) t_0)
       (+ (cos phi1) t_1)))
     (+
      lambda1
      (atan2
       (* (cos (fabs phi2)) t_0)
       (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) t_1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (fabs(phi2) <= 31000000000.0) {
		tmp = lambda1 + atan2(((1.0 + (-0.5 * pow(fabs(phi2), 2.0))) * t_0), (cos(phi1) + t_1));
	} else {
		tmp = lambda1 + atan2((cos(fabs(phi2)) * t_0), ((1.0 + (-0.5 * pow(phi1, 2.0))) + t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = cos((lambda1 - lambda2))
    if (abs(phi2) <= 31000000000.0d0) then
        tmp = lambda1 + atan2(((1.0d0 + ((-0.5d0) * (abs(phi2) ** 2.0d0))) * t_0), (cos(phi1) + t_1))
    else
        tmp = lambda1 + atan2((cos(abs(phi2)) * t_0), ((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) + t_1))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (Math.abs(phi2) <= 31000000000.0) {
		tmp = lambda1 + Math.atan2(((1.0 + (-0.5 * Math.pow(Math.abs(phi2), 2.0))) * t_0), (Math.cos(phi1) + t_1));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(Math.abs(phi2)) * t_0), ((1.0 + (-0.5 * Math.pow(phi1, 2.0))) + t_1));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if math.fabs(phi2) <= 31000000000.0:
		tmp = lambda1 + math.atan2(((1.0 + (-0.5 * math.pow(math.fabs(phi2), 2.0))) * t_0), (math.cos(phi1) + t_1))
	else:
		tmp = lambda1 + math.atan2((math.cos(math.fabs(phi2)) * t_0), ((1.0 + (-0.5 * math.pow(phi1, 2.0))) + t_1))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (abs(phi2) <= 31000000000.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(1.0 + Float64(-0.5 * (abs(phi2) ^ 2.0))) * t_0), Float64(cos(phi1) + t_1)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(abs(phi2)) * t_0), Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (abs(phi2) <= 31000000000.0)
		tmp = lambda1 + atan2(((1.0 + (-0.5 * (abs(phi2) ^ 2.0))) * t_0), (cos(phi1) + t_1));
	else
		tmp = lambda1 + atan2((cos(abs(phi2)) * t_0), ((1.0 + (-0.5 * (phi1 ^ 2.0))) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 31000000000.0], N[(lambda1 + N[ArcTan[N[(N[(1.0 + N[(-0.5 * N[Power[N[Abs[phi2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.1e10

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 74.9

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 74.9%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 3.1e10 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 68.8%

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

Alternative 17: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\left|\phi_2\right| \cdot \left|\phi_2\right|, -0.5, 1\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 3.1:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\mathsf{fma}\left(t\_1, \cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \left(\left|\phi_2\right|\right) \cdot t\_0}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (fma (* (fabs phi2) (fabs phi2)) -0.5 1.0)))
   (if (<= (fabs phi2) 3.1)
     (+
      (atan2 (* t_1 t_0) (fma t_1 (cos (- lambda2 lambda1)) (cos phi1)))
      lambda1)
     (+
      lambda1
      (atan2
       (* (cos (fabs phi2)) t_0)
       (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma((fabs(phi2) * fabs(phi2)), -0.5, 1.0);
	double tmp;
	if (fabs(phi2) <= 3.1) {
		tmp = atan2((t_1 * t_0), fma(t_1, cos((lambda2 - lambda1)), cos(phi1))) + lambda1;
	} else {
		tmp = lambda1 + atan2((cos(fabs(phi2)) * t_0), ((1.0 + (-0.5 * pow(phi1, 2.0))) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(abs(phi2) * abs(phi2)), -0.5, 1.0)
	tmp = 0.0
	if (abs(phi2) <= 3.1)
		tmp = Float64(atan(Float64(t_1 * t_0), fma(t_1, cos(Float64(lambda2 - lambda1)), cos(phi1))) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(cos(abs(phi2)) * t_0), Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[phi2], $MachinePrecision] * N[Abs[phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 3.1], N[(N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(t$95$1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.10000000000000009

   1. Initial program 98.6%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. sin-neg N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lower-PI.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Applied rewrites 51.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.4

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   9. Applied rewrites 52.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 52.4

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1} \]

   11. Applied rewrites 75.9%

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

   if 3.10000000000000009 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 68.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 68.8%

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

Alternative 18: 77.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \mathsf{fma}\left(\left|\phi_2\right| \cdot \left|\phi_2\right|, -0.5, 1\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 7.6 \cdot 10^{+48}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2 \cdot t\_0}{\mathsf{fma}\left(t\_2, t\_1, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \left(\left|\phi_2\right|\right)}{t\_1 - -1} + \lambda_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (fma (* (fabs phi2) (fabs phi2)) -0.5 1.0)))
   (if (<= (fabs phi2) 7.6e+48)
     (+ (atan2 (* t_2 t_0) (fma t_2 t_1 (cos phi1))) lambda1)
     (+ (atan2 (* t_0 (cos (fabs phi2))) (- t_1 -1.0)) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = fma((fabs(phi2) * fabs(phi2)), -0.5, 1.0);
	double tmp;
	if (fabs(phi2) <= 7.6e+48) {
		tmp = atan2((t_2 * t_0), fma(t_2, t_1, cos(phi1))) + lambda1;
	} else {
		tmp = atan2((t_0 * cos(fabs(phi2))), (t_1 - -1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = fma(Float64(abs(phi2) * abs(phi2)), -0.5, 1.0)
	tmp = 0.0
	if (abs(phi2) <= 7.6e+48)
		tmp = Float64(atan(Float64(t_2 * t_0), fma(t_2, t_1, cos(phi1))) + lambda1);
	else
		tmp = Float64(atan(Float64(t_0 * cos(abs(phi2))), Float64(t_1 - -1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[phi2], $MachinePrecision] * N[Abs[phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 7.6e+48], N[(N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(t$95$2 * t$95$1 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(t$95$0 * N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.60000000000000001e48

   1. Initial program 98.6%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. sin-neg N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lower-PI.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Applied rewrites 51.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.4

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   9. Applied rewrites 52.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 52.4

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) + \pi\right)}{\cos \phi_1 + \left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1} \]

   11. Applied rewrites 75.9%

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

   if 7.60000000000000001e48 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 67.3

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

   9. Applied rewrites 67.3%

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

Alternative 19: 67.3% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  (atan2
   (* (sin (- lambda1 lambda2)) (cos phi2))
   (- (cos (- lambda2 lambda1)) -1.0))
  lambda1))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), (cos((lambda2 - lambda1)) - -1.0)) + lambda1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (cos((lambda2 - lambda1)) - (-1.0d0))) + lambda1
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (Math.cos((lambda2 - lambda1)) - -1.0)) + lambda1;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (math.cos((lambda2 - lambda1)) - -1.0)) + lambda1

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(cos(Float64(lambda2 - lambda1)) - -1.0)) + lambda1)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (cos((lambda2 - lambda1)) - -1.0)) + lambda1;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 67.3

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

7. Applied rewrites 67.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 67.3

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

9. Applied rewrites 67.3%

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

Alternative 20: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\phi_2\right| \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\left|\phi_2\right| \cdot \left|\phi_2\right|, -0.5, 1\right)}{\cos \left(\lambda_2 - \lambda_1\right) - -1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \left(\left|\phi_2\right|\right) \cdot \sin \left(-\lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fabs phi2) 1.15e-15)
   (+
    (atan2
     (* (sin (- lambda1 lambda2)) (fma (* (fabs phi2) (fabs phi2)) -0.5 1.0))
     (- (cos (- lambda2 lambda1)) -1.0))
    lambda1)
   (+
    lambda1
    (atan2
     (* (cos (fabs phi2)) (sin (- lambda2)))
     (+ 1.0 (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(phi2) <= 1.15e-15) {
		tmp = atan2((sin((lambda1 - lambda2)) * fma((fabs(phi2) * fabs(phi2)), -0.5, 1.0)), (cos((lambda2 - lambda1)) - -1.0)) + lambda1;
	} else {
		tmp = lambda1 + atan2((cos(fabs(phi2)) * sin(-lambda2)), (1.0 + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(phi2) <= 1.15e-15)
		tmp = Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(abs(phi2) * abs(phi2)), -0.5, 1.0)), Float64(cos(Float64(lambda2 - lambda1)) - -1.0)) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(cos(abs(phi2)) * sin(Float64(-lambda2))), Float64(1.0 + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Abs[phi2], $MachinePrecision], 1.15e-15], N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Abs[phi2], $MachinePrecision] * N[Abs[phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.14999999999999995e-15

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 64.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.8

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1} \]

   12. Applied rewrites 64.8%

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

   if 1.14999999999999995e-15 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in lambda1 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-neg.f64 64.1

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

   10. Applied rewrites 64.1%

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

Alternative 21: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 1.2:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{fma}\left(\left|\phi_2\right| \cdot \left|\phi_2\right|, -0.5, 1\right)}{\cos \left(\lambda_2 - \lambda_1\right) - -1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \left(\left|\phi_2\right|\right) \cdot t\_0}{1 + \cos \lambda_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (fabs phi2) 1.2)
     (+
      (atan2
       (* t_0 (fma (* (fabs phi2) (fabs phi2)) -0.5 1.0))
       (- (cos (- lambda2 lambda1)) -1.0))
      lambda1)
     (+ lambda1 (atan2 (* (cos (fabs phi2)) t_0) (+ 1.0 (cos lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (fabs(phi2) <= 1.2) {
		tmp = atan2((t_0 * fma((fabs(phi2) * fabs(phi2)), -0.5, 1.0)), (cos((lambda2 - lambda1)) - -1.0)) + lambda1;
	} else {
		tmp = lambda1 + atan2((cos(fabs(phi2)) * t_0), (1.0 + cos(lambda1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (abs(phi2) <= 1.2)
		tmp = Float64(atan(Float64(t_0 * fma(Float64(abs(phi2) * abs(phi2)), -0.5, 1.0)), Float64(cos(Float64(lambda2 - lambda1)) - -1.0)) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(cos(abs(phi2)) * t_0), Float64(1.0 + cos(lambda1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 1.2], N[(N[ArcTan[N[(t$95$0 * N[(N[(N[Abs[phi2], $MachinePrecision] * N[Abs[phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.19999999999999996

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 64.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.8

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1} \]

   12. Applied rewrites 64.8%

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

   if 1.19999999999999996 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower--.f64 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 62.0

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

   10. Applied rewrites 62.0%

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

Alternative 22: 64.8% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  (atan2
   (* (sin (- lambda1 lambda2)) (fma (* phi2 phi2) -0.5 1.0))
   (- (cos (- lambda2 lambda1)) -1.0))
  lambda1))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * fma((phi2 * phi2), -0.5, 1.0)), (cos((lambda2 - lambda1)) - -1.0)) + lambda1;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(cos(Float64(lambda2 - lambda1)) - -1.0)) + lambda1)
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 67.3

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

7. Applied rewrites 67.3%

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

8. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 64.8

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

10. Applied rewrites 64.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
11. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 64.8

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(1 + -0.5 \cdot {\phi_2}^{2}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1} \]

12. Applied rewrites 64.8%

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

Reproduce

herbie shell --seed 2025168 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))