Midpoint on a great circle

Percentage Accurate: 98.6% → 98.6%

Time: 10.4s

Alternatives: 14

Speedup: 0.7×

Specification

Math

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

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

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

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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\sin phi2\_m, \cos \left(\frac{\pi}{2}\right), \cos phi2\_m \cdot 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2_m) (sin (- lambda1 lambda2)))
   (+
    (cos phi1)
    (*
     (fma (sin phi2_m) (cos (/ PI 2.0)) (* (cos phi2_m) 1.0))
     (cos (- lambda1 lambda2)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + atan2((cos(phi2_m) * sin((lambda1 - lambda2))), (cos(phi1) + (fma(sin(phi2_m), cos((((double) M_PI) / 2.0)), (cos(phi2_m) * 1.0)) * cos((lambda1 - lambda2)))));
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(fma(sin(phi2_m), cos(Float64(pi / 2.0)), Float64(cos(phi2_m) * 1.0)) * cos(Float64(lambda1 - lambda2))))))
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[(N[Sin[phi2$95$m], $MachinePrecision] * N[Cos[N[(Pi / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[phi2$95$m], $MachinePrecision] * 1.0), $MachinePrecision]), $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-cos.f64 N/A

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

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

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

   3. sin-sum N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sin-PI/2 98.6

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

5. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos phi2\_m \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2_m) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2_m) (cos (- lambda1 lambda2)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + atan2((cos(phi2_m) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2_m) * cos((lambda1 - lambda2)))));
}

Fortran

phi2_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2_m
    code = lambda1 + atan2((cos(phi2_m) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2_m) * cos((lambda1 - lambda2)))))
end function

Java

phi2_m = Math.abs(phi2);
public static double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + Math.atan2((Math.cos(phi2_m) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2_m) * Math.cos((lambda1 - lambda2)))));
}

Python

phi2_m = math.fabs(phi2)
def code(lambda1, lambda2, phi1, phi2_m):
	return lambda1 + math.atan2((math.cos(phi2_m) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2_m) * math.cos((lambda1 - lambda2)))))

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2_m) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

phi2_m = abs(phi2);
function tmp = code(lambda1, lambda2, phi1, phi2_m)
	tmp = lambda1 + atan2((cos(phi2_m) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2_m) * cos((lambda1 - lambda2)))));
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2$95$m], $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. Add Preprocessing

Alternative 3: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos phi2\_m \cdot t\_0\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \lambda_1, \cos phi2\_m, \cos \phi_1\right)}\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos phi2\_m \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_4 := \tan^{-1}_* \frac{t\_0 \cdot \cos phi2\_m}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos phi2\_m, \cos \phi_1\right)}\\ \mathbf{if}\;t\_3 \leq -50:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.08:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 3:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (cos phi2_m) t_0))
        (t_2
         (+ lambda1 (atan2 t_1 (fma (cos lambda1) (cos phi2_m) (cos phi1)))))
        (t_3
         (+
          lambda1
          (atan2
           t_1
           (+ (cos phi1) (* (cos phi2_m) (cos (- lambda1 lambda2)))))))
        (t_4
         (atan2
          (* t_0 (cos phi2_m))
          (fma (cos (- lambda2 lambda1)) (cos phi2_m) (cos phi1)))))
   (if (<= t_3 -50.0)
     t_2
     (if (<= t_3 -0.08)
       t_4
       (if (<= t_3 5e-7) t_2 (if (<= t_3 3.0) t_4 t_2))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi2_m) * t_0;
	double t_2 = lambda1 + atan2(t_1, fma(cos(lambda1), cos(phi2_m), cos(phi1)));
	double t_3 = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2_m) * cos((lambda1 - lambda2)))));
	double t_4 = atan2((t_0 * cos(phi2_m)), fma(cos((lambda2 - lambda1)), cos(phi2_m), cos(phi1)));
	double tmp;
	if (t_3 <= -50.0) {
		tmp = t_2;
	} else if (t_3 <= -0.08) {
		tmp = t_4;
	} else if (t_3 <= 5e-7) {
		tmp = t_2;
	} else if (t_3 <= 3.0) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2_m) * t_0)
	t_2 = Float64(lambda1 + atan(t_1, fma(cos(lambda1), cos(phi2_m), cos(phi1))))
	t_3 = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2_m) * cos(Float64(lambda1 - lambda2))))))
	t_4 = atan(Float64(t_0 * cos(phi2_m)), fma(cos(Float64(lambda2 - lambda1)), cos(phi2_m), cos(phi1)))
	tmp = 0.0
	if (t_3 <= -50.0)
		tmp = t_2;
	elseif (t_3 <= -0.08)
		tmp = t_4;
	elseif (t_3 <= 5e-7)
		tmp = t_2;
	elseif (t_3 <= 3.0)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2$95$m], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(t$95$0 * N[Cos[phi2$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -50.0], t$95$2, If[LessEqual[t$95$3, -0.08], t$95$4, If[LessEqual[t$95$3, 5e-7], t$95$2, If[LessEqual[t$95$3, 3.0], t$95$4, t$95$2]]]]]]]]]

Derivation

1. Split input into 2 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)))))) < -50 or -0.0800000000000000017 < (+.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)))))) < 4.99999999999999977e-7 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 99.0%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if -50 < (+.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)))))) < -0.0800000000000000017 or 4.99999999999999977e-7 < (+.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 97.5%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. negate-sub2 N/A

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

      11. cos-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-cos.f64 96.1

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

   4. Applied rewrites 96.1%

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

Alternative 4: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \lambda_1}{\mathsf{fma}\left(\cos \lambda_1, \cos phi2\_m, \cos \phi_1\right)}\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos phi2\_m \cdot t\_1\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos \phi_1 + \cos phi2\_m \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_4 := \tan^{-1}_* \frac{t\_1 \cdot \cos phi2\_m}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos phi2\_m, \cos \phi_1\right)}\\ \mathbf{if}\;t\_3 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -0.04:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos \phi_1 + \cos phi2\_m}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (let* ((t_0
         (+
          lambda1
          (atan2
           (* (cos phi2_m) (sin lambda1))
           (fma (cos lambda1) (cos phi2_m) (cos phi1)))))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2_m) t_1))
        (t_3
         (+
          lambda1
          (atan2
           t_2
           (+ (cos phi1) (* (cos phi2_m) (cos (- lambda1 lambda2)))))))
        (t_4
         (atan2
          (* t_1 (cos phi2_m))
          (fma (cos (- lambda2 lambda1)) (cos phi2_m) (cos phi1)))))
   (if (<= t_3 -50.0)
     t_0
     (if (<= t_3 -0.04)
       t_4
       (if (<= t_3 5e-7)
         (+ lambda1 (atan2 t_2 (+ (cos phi1) (cos phi2_m))))
         (if (<= t_3 5.0) t_4 t_0))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double t_0 = lambda1 + atan2((cos(phi2_m) * sin(lambda1)), fma(cos(lambda1), cos(phi2_m), cos(phi1)));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = cos(phi2_m) * t_1;
	double t_3 = lambda1 + atan2(t_2, (cos(phi1) + (cos(phi2_m) * cos((lambda1 - lambda2)))));
	double t_4 = atan2((t_1 * cos(phi2_m)), fma(cos((lambda2 - lambda1)), cos(phi2_m), cos(phi1)));
	double tmp;
	if (t_3 <= -50.0) {
		tmp = t_0;
	} else if (t_3 <= -0.04) {
		tmp = t_4;
	} else if (t_3 <= 5e-7) {
		tmp = lambda1 + atan2(t_2, (cos(phi1) + cos(phi2_m)));
	} else if (t_3 <= 5.0) {
		tmp = t_4;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	t_0 = Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(lambda1)), fma(cos(lambda1), cos(phi2_m), cos(phi1))))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2_m) * t_1)
	t_3 = Float64(lambda1 + atan(t_2, Float64(cos(phi1) + Float64(cos(phi2_m) * cos(Float64(lambda1 - lambda2))))))
	t_4 = atan(Float64(t_1 * cos(phi2_m)), fma(cos(Float64(lambda2 - lambda1)), cos(phi2_m), cos(phi1)))
	tmp = 0.0
	if (t_3 <= -50.0)
		tmp = t_0;
	elseif (t_3 <= -0.04)
		tmp = t_4;
	elseif (t_3 <= 5e-7)
		tmp = Float64(lambda1 + atan(t_2, Float64(cos(phi1) + cos(phi2_m))));
	elseif (t_3 <= 5.0)
		tmp = t_4;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := Block[{t$95$0 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(t$95$1 * N[Cos[phi2$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -50.0], t$95$0, If[LessEqual[t$95$3, -0.04], t$95$4, If[LessEqual[t$95$3, 5e-7], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[phi2$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], t$95$4, t$95$0]]]]]]]]]

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)))))) < -50 or 5 < (+.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 99.0%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in lambda1 around inf

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

   7. Applied rewrites 99.0%

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

   if -50 < (+.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)))))) < -0.0400000000000000008 or 4.99999999999999977e-7 < (+.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)))))) < 5

   1. Initial program 97.7%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. negate-sub2 N/A

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

      11. cos-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-cos.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if -0.0400000000000000008 < (+.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)))))) < 4.99999999999999977e-7

   1. Initial program 99.0%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \lambda_2, \cos phi2\_m, \cos \phi_1\right)} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2_m) (sin (- lambda1 lambda2)))
   (fma (cos lambda2) (cos phi2_m) (cos phi1)))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + atan2((cos(phi2_m) * sin((lambda1 - lambda2))), fma(cos(lambda2), cos(phi2_m), cos(phi1)));
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(lambda1 - lambda2))), fma(cos(lambda2), cos(phi2_m), cos(phi1))))
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + N[Cos[phi1], $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 6: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} t_0 := \cos phi2\_m \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 9 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos phi2\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \mathsf{fma}\left(phi2\_m \cdot phi2\_m, -0.5, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (let* ((t_0 (* (cos phi2_m) (sin (- lambda1 lambda2)))))
   (if (<= phi1 9e-22)
     (+ lambda1 (atan2 t_0 (fma (cos (- lambda2 lambda1)) (cos phi2_m) 1.0)))
     (+
      lambda1
      (atan2
       t_0
       (+
        (cos phi1)
        (* (fma (* phi2_m phi2_m) -0.5 1.0) (cos (- lambda1 lambda2)))))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double t_0 = cos(phi2_m) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 9e-22) {
		tmp = lambda1 + atan2(t_0, fma(cos((lambda2 - lambda1)), cos(phi2_m), 1.0));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (fma((phi2_m * phi2_m), -0.5, 1.0) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	t_0 = Float64(cos(phi2_m) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 9e-22)
		tmp = Float64(lambda1 + atan(t_0, fma(cos(Float64(lambda2 - lambda1)), cos(phi2_m), 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(fma(Float64(phi2_m * phi2_m), -0.5, 1.0) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := Block[{t$95$0 = N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 9e-22], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2$95$m], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[(N[(phi2$95$m * phi2$95$m), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 8.99999999999999973e-22

   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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub2 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if 8.99999999999999973e-22 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.7

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

   4. Applied rewrites 82.7%

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

Alternative 7: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos phi2\_m \leq 0.999995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot t\_0}{\cos \phi_1 + \cos phi2\_m}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2_m) 0.999995)
     (+ lambda1 (atan2 (* (cos phi2_m) t_0) (+ (cos phi1) (cos phi2_m))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda2 lambda1))))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2_m) <= 0.999995) {
		tmp = lambda1 + atan2((cos(phi2_m) * t_0), (cos(phi1) + cos(phi2_m)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

phi2_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (cos(phi2_m) <= 0.999995d0) then
        tmp = lambda1 + atan2((cos(phi2_m) * t_0), (cos(phi1) + cos(phi2_m)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

phi2_m = Math.abs(phi2);
public static double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2_m) <= 0.999995) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2_m) * t_0), (Math.cos(phi1) + Math.cos(phi2_m)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

phi2_m = math.fabs(phi2)
def code(lambda1, lambda2, phi1, phi2_m):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2_m) <= 0.999995:
		tmp = lambda1 + math.atan2((math.cos(phi2_m) * t_0), (math.cos(phi1) + math.cos(phi2_m)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos((lambda2 - lambda1))))
	return tmp

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2_m) <= 0.999995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2_m) * t_0), Float64(cos(phi1) + cos(phi2_m))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

phi2_m = abs(phi2);
function tmp_2 = code(lambda1, lambda2, phi1, phi2_m)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2_m) <= 0.999995)
		tmp = lambda1 + atan2((cos(phi2_m) * t_0), (cos(phi1) + cos(phi2_m)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2$95$m], $MachinePrecision], 0.999995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[phi2$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.99999499999999997

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if 0.99999499999999997 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-neg-rev N/A

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

      2. negate-sub2 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 8: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} \mathbf{if}\;\cos phi2\_m \leq 0.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(\left(1 - \frac{\lambda_2}{\lambda_1}\right) \cdot \lambda_1\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + 1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (if (<= (cos phi2_m) 0.5)
   (+
    lambda1
    (atan2
     (* (cos phi2_m) (sin (* (- 1.0 (/ lambda2 lambda1)) lambda1)))
     (+ (fma (* phi1 phi1) -0.5 1.0) (* 1.0 (cos (- lambda1 lambda2))))))
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (+ (cos phi1) (cos (- lambda2 lambda1)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double tmp;
	if (cos(phi2_m) <= 0.5) {
		tmp = lambda1 + atan2((cos(phi2_m) * sin(((1.0 - (lambda2 / lambda1)) * lambda1))), (fma((phi1 * phi1), -0.5, 1.0) + (1.0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	tmp = 0.0
	if (cos(phi2_m) <= 0.5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(Float64(1.0 - Float64(lambda2 / lambda1)) * lambda1))), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(1.0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := If[LessEqual[N[Cos[phi2$95$m], $MachinePrecision], 0.5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(N[(1.0 - N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(1.0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.5

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. frac-2neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if 0.5 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 87.6

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-neg-rev N/A

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

      2. negate-sub2 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 89.2

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

   10. Applied rewrites 89.2%

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

Alternative 9: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} \mathbf{if}\;\cos phi2\_m \leq 0.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(\left(1 - \frac{\lambda_2}{\lambda_1}\right) \cdot \lambda_1\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + 1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (if (<= (cos phi2_m) 0.5)
   (+
    lambda1
    (atan2
     (* (cos phi2_m) (sin (* (- 1.0 (/ lambda2 lambda1)) lambda1)))
     (+ (fma (* phi1 phi1) -0.5 1.0) (* 1.0 (cos lambda1)))))
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (+ (cos phi1) (cos (- lambda2 lambda1)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double tmp;
	if (cos(phi2_m) <= 0.5) {
		tmp = lambda1 + atan2((cos(phi2_m) * sin(((1.0 - (lambda2 / lambda1)) * lambda1))), (fma((phi1 * phi1), -0.5, 1.0) + (1.0 * cos(lambda1))));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	tmp = 0.0
	if (cos(phi2_m) <= 0.5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(Float64(1.0 - Float64(lambda2 / lambda1)) * lambda1))), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(1.0 * cos(lambda1)))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := If[LessEqual[N[Cos[phi2$95$m], $MachinePrecision], 0.5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[N[(N[(1.0 - N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(1.0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.5

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. frac-2neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   11. Taylor expanded in lambda1 around inf

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

   13. Applied rewrites 63.2%

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

   if 0.5 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 87.6

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-neg-rev N/A

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

      2. negate-sub2 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 89.2

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

   10. Applied rewrites 89.2%

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

Alternative 10: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} \mathbf{if}\;\cos phi2\_m \leq -0.05:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \left(-\lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + 1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (if (<= (cos phi2_m) -0.05)
   (+
    lambda1
    (atan2
     (* (cos phi2_m) (sin (- lambda2)))
     (+ (fma (* phi1 phi1) -0.5 1.0) (* 1.0 (cos (- lambda1 lambda2))))))
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (+ (cos phi1) (cos (- lambda2 lambda1)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double tmp;
	if (cos(phi2_m) <= -0.05) {
		tmp = lambda1 + atan2((cos(phi2_m) * sin(-lambda2)), (fma((phi1 * phi1), -0.5, 1.0) + (1.0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	tmp = 0.0
	if (cos(phi2_m) <= -0.05)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(Float64(-lambda2))), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(1.0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := If[LessEqual[N[Cos[phi2$95$m], $MachinePrecision], -0.05], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(1.0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.050000000000000003

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. frac-2neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 54.9%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.9

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

   10. Applied rewrites 63.9%

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

   11. Taylor expanded in lambda1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 64.0

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

   13. Applied rewrites 64.0%

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

   if -0.050000000000000003 < (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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 84.2

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-neg-rev N/A

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

      2. negate-sub2 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 85.9

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

   10. Applied rewrites 85.9%

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

Alternative 11: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \begin{array}{l} \mathbf{if}\;\cos phi2\_m \leq 0.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos phi2\_m \cdot \sin \lambda_1}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + 1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (if (<= (cos phi2_m) 0.5)
   (+
    lambda1
    (atan2
     (* (cos phi2_m) (sin lambda1))
     (+ (fma (* phi1 phi1) -0.5 1.0) (* 1.0 (cos (- lambda1 lambda2))))))
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (+ (cos phi1) (cos (- lambda2 lambda1)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	double tmp;
	if (cos(phi2_m) <= 0.5) {
		tmp = lambda1 + atan2((cos(phi2_m) * sin(lambda1)), (fma((phi1 * phi1), -0.5, 1.0) + (1.0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	tmp = 0.0
	if (cos(phi2_m) <= 0.5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2_m) * sin(lambda1)), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(1.0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := If[LessEqual[N[Cos[phi2$95$m], $MachinePrecision], 0.5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2$95$m], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(1.0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.5

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. frac-2neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   11. Taylor expanded in lambda1 around inf

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

   13. Applied rewrites 60.0%

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

   if 0.5 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. negate-sub N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 87.6

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in phi2 around 0

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

      1. cos-neg-rev N/A

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

      2. negate-sub2 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 89.2

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

   10. Applied rewrites 89.2%

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

Alternative 12: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (+
  lambda1
  (atan2 (sin (- lambda1 lambda2)) (+ (cos phi1) (cos (- lambda2 lambda1))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
}

Fortran

phi2_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2_m
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))))
end function

Java

phi2_m = Math.abs(phi2);
public static double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
}

Python

phi2_m = math.fabs(phi2)
def code(lambda1, lambda2, phi1, phi2_m):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(phi1) + math.cos((lambda2 - lambda1))))

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))))
end

MATLAB

phi2_m = abs(phi2);
function tmp = code(lambda1, lambda2, phi1, phi2_m)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda2 - lambda1))));
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $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. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. negate-sub N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 77.1

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

4. Applied rewrites 77.1%

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

5. Taylor expanded in phi2 around 0

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 76.4

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

7. Applied rewrites 76.4%

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

8. Taylor expanded in phi2 around 0

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

   1. cos-neg-rev N/A

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

   2. negate-sub2 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 77.6

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

10. Applied rewrites 77.6%

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

Alternative 13: 67.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + 1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m)
 :precision binary64
 (+
  lambda1
  (atan2 (sin (- lambda1 lambda2)) (+ 1.0 (* 1.0 (cos (- lambda1 lambda2)))))))

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (1.0 + (1.0 * cos((lambda1 - lambda2)))));
}

Fortran

phi2_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2_m
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (1.0d0 + (1.0d0 * cos((lambda1 - lambda2)))))
end function

Java

phi2_m = Math.abs(phi2);
public static double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (1.0 + (1.0 * Math.cos((lambda1 - lambda2)))));
}

Python

phi2_m = math.fabs(phi2)
def code(lambda1, lambda2, phi1, phi2_m):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (1.0 + (1.0 * math.cos((lambda1 - lambda2)))))

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(1.0 + Float64(1.0 * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

phi2_m = abs(phi2);
function tmp = code(lambda1, lambda2, phi1, phi2_m)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (1.0 + (1.0 * cos((lambda1 - lambda2)))));
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(1.0 * 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. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. negate-sub N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 77.1

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

4. Applied rewrites 77.1%

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

5. Taylor expanded in phi2 around 0

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 76.4

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

7. Applied rewrites 76.4%

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

8. Taylor expanded in phi1 around 0

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

10. Applied rewrites 66.4%

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

11. Taylor expanded in phi2 around 0

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

13. Applied rewrites 67.3%

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

Alternative 14: 53.0% accurate, 192.4× speedup

Math

\[\begin{array}{l} phi2_m = \left|\phi_2\right| \\ \lambda_1 \end{array} \]

FPCore

phi2_m = (fabs.f64 phi2)
(FPCore (lambda1 lambda2 phi1 phi2_m) :precision binary64 lambda1)

C

phi2_m = fabs(phi2);
double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1;
}

Fortran

phi2_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2_m
    code = lambda1
end function

Java

phi2_m = Math.abs(phi2);
public static double code(double lambda1, double lambda2, double phi1, double phi2_m) {
	return lambda1;
}

Python

phi2_m = math.fabs(phi2)
def code(lambda1, lambda2, phi1, phi2_m):
	return lambda1

Julia

phi2_m = abs(phi2)
function code(lambda1, lambda2, phi1, phi2_m)
	return lambda1
end

MATLAB

phi2_m = abs(phi2);
function tmp = code(lambda1, lambda2, phi1, phi2_m)
	tmp = lambda1;
end

Wolfram

phi2_m = N[Abs[phi2], $MachinePrecision]
code[lambda1_, lambda2_, phi1_, phi2$95$m_] := lambda1

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 lambda1 around inf

   \[\leadsto \color{blue}{\lambda_1} \]
3. Step-by-step derivation

4. Applied rewrites 53.0%

   \[\leadsto \color{blue}{\lambda_1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025118 
(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)))))))