Midpoint on a great circle

Percentage Accurate: 98.5% → 99.6%

Time: 11.1s

Alternatives: 13

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

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

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

   5. lower--.f64 99.6%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(lambda1) * cos(lambda2)
    code = lambda1 + atan2((cos(phi2) * ((1.0d0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(fma(cos(Float64(lambda2 - lambda1)), cos(phi2), cos(phi1)) * Float64(1.0 / cos(phi1))) * cos(phi1))))
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[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. sum-to-mult N/A

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

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

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

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

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

   5. lower-unsound-/.f64 98.5%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.5%

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

   9. lift-cos.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f64 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. mult-flip N/A

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

   5. *-lft-identity N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 98.5%

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

5. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in lambda1 around 0

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

6. Applied rewrites 97.8%

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

Alternative 6: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{1 + \mathsf{fma}\left(-0.5, {\phi_1}^{2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0, \cos \phi_1\right)} + \lambda_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma (fma 0.041666666666666664 (* phi2 phi2) -0.5) (* phi2 phi2) 1.0))
        (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.995)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (+
        1.0
        (fma -0.5 (pow phi1 2.0) (* (cos phi2) (cos (- lambda1 lambda2)))))))
     (+
      (atan2 (* t_1 t_0) (fma (cos (- lambda2 lambda1)) t_0 (cos phi1)))
      lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(fma(0.041666666666666664, (phi2 * phi2), -0.5), (phi2 * phi2), 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.995) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), (1.0 + fma(-0.5, pow(phi1, 2.0), (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((t_1 * t_0), fma(cos((lambda2 - lambda1)), t_0, cos(phi1))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(fma(0.041666666666666664, Float64(phi2 * phi2), -0.5), Float64(phi2 * phi2), 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(1.0 + fma(-0.5, (phi1 ^ 2.0), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(atan(Float64(t_1 * t_0), fma(cos(Float64(lambda2 - lambda1)), t_0, cos(phi1))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(phi2 * phi2), $MachinePrecision] + -0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.994999999999999996

   1. Initial program 98.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 80.2%

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

   4. Applied rewrites 80.2%

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

   if 0.994999999999999996 < (cos.f64 phi2)

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.5%

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 75.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, \cos \phi_1\right)} + \lambda_1 \]

   6. Applied rewrites 75.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)\right)}}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, \cos \phi_1\right)} + \lambda_1 \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)}, \cos \phi_1\right)} + \lambda_1 \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 75.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]

   9. Applied rewrites 75.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right)}, \cos \phi_1\right)} + \lambda_1 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 75.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5, \color{blue}{{\phi_2}^{2}}, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]

       6. lift--.f64 N/A

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

       7. sub-flip N/A

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

       8. lift-*.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 75.9%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, {\phi_2}^{2}, -0.5\right), {\color{blue}{\phi_2}}^{2}, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]

       11. lift-pow.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 75.9%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), {\phi_2}^{2}, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]

       14. lift-pow.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 75.9%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \color{blue}{\phi_2}, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]

   11. Applied rewrites 75.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 1 + {\phi_2}^{2} \cdot \left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5\right), \cos \phi_1\right)} + \lambda_1 \]
   12. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 75.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \mathsf{fma}\left(0.041666666666666664 \cdot {\phi_2}^{2} - 0.5, \color{blue}{{\phi_2}^{2}}, 1\right), \cos \phi_1\right)} + \lambda_1 \]

       6. lift--.f64 N/A

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

       7. sub-flip N/A

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

       8. lift-*.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 75.9%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, {\phi_2}^{2}, -0.5\right), {\color{blue}{\phi_2}}^{2}, 1\right), \cos \phi_1\right)} + \lambda_1 \]

       11. lift-pow.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 75.9%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right), {\phi_2}^{2}, 1\right), \cos \phi_1\right)} + \lambda_1 \]

       14. lift-pow.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 75.9%

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

   13. Applied rewrites 75.9%

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

Alternative 7: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.992:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0)) (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.992)
     (+
      (atan2 (* t_1 t_0) (fma (cos (- lambda2 lambda1)) t_0 (cos phi1)))
      lambda1)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (+ 1.0 (* (cos phi2) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.992) {
		tmp = atan2((t_1 * t_0), fma(cos((lambda2 - lambda1)), t_0, cos(phi1))) + lambda1;
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_1), (1.0 + (cos(phi2) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.992)
		tmp = Float64(atan(Float64(t_1 * t_0), fma(cos(Float64(lambda2 - lambda1)), t_0, cos(phi1))) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(1.0 + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.992], N[(N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99199999999999999

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.5%

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.2%

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.9%

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

   9. Applied rewrites 76.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 76.9%

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 76.9%

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

   11. Applied rewrites 76.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 76.9%

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 76.9%

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

   13. Applied rewrites 76.9%

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

   if 0.99199999999999999 < (cos.f64 phi1)

   1. Initial program 98.5%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 78.5%

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

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.97:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.97)
     (+ lambda1 (atan2 t_1 (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) t_0)))
     (+ lambda1 (atan2 t_1 (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.97) {
		tmp = lambda1 + atan2(t_1, ((1.0 + (-0.5 * pow(phi1, 2.0))) + t_0));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.97d0) then
        tmp = lambda1 + atan2(t_1, ((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) + t_0))
    else
        tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.97) {
		tmp = lambda1 + Math.atan2(t_1, ((1.0 + (-0.5 * Math.pow(phi1, 2.0))) + t_0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.97:
		tmp = lambda1 + math.atan2(t_1, ((1.0 + (-0.5 * math.pow(phi1, 2.0))) + t_0))
	else:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.97)
		tmp = Float64(lambda1 + atan(t_1, Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + t_0)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.97)
		tmp = lambda1 + atan2(t_1, ((1.0 + (-0.5 * (phi1 ^ 2.0))) + t_0));
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.97], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.96999999999999997

   1. Initial program 98.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 78.4%

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 70.1%

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

   7. Applied rewrites 70.1%

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

   if 0.96999999999999997 < (cos.f64 phi2)

   1. Initial program 98.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 78.4%

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

   4. Applied rewrites 78.4%

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

Alternative 9: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\left|\phi_2\right| \cdot \left|\phi_2\right|, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\left|\phi_2\right| \leq 1.8:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \left(\left|\phi_2\right|\right) \cdot t\_1}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* (fabs phi2) (fabs phi2)) -0.5 1.0))
        (t_1 (sin (- lambda1 lambda2))))
   (if (<= (fabs phi2) 1.8)
     (+
      (atan2 (* t_1 t_0) (fma (cos (- lambda2 lambda1)) t_0 (cos phi1)))
      lambda1)
     (+
      lambda1
      (atan2
       (* (cos (fabs phi2)) t_1)
       (+ (+ 1.0 (* -0.5 (pow phi1 2.0))) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((fabs(phi2) * fabs(phi2)), -0.5, 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (fabs(phi2) <= 1.8) {
		tmp = atan2((t_1 * t_0), fma(cos((lambda2 - lambda1)), t_0, cos(phi1))) + lambda1;
	} else {
		tmp = lambda1 + atan2((cos(fabs(phi2)) * t_1), ((1.0 + (-0.5 * pow(phi1, 2.0))) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(abs(phi2) * abs(phi2)), -0.5, 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (abs(phi2) <= 1.8)
		tmp = Float64(atan(Float64(t_1 * t_0), fma(cos(Float64(lambda2 - lambda1)), t_0, cos(phi1))) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(cos(abs(phi2)) * t_1), Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Abs[phi2], $MachinePrecision] * N[Abs[phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[phi2], $MachinePrecision], 1.8], N[(N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[N[Abs[phi2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.80000000000000004

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.5%

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.2%

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.9%

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

   9. Applied rewrites 76.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 76.9%

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 76.9%

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

   11. Applied rewrites 76.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 76.9%

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 76.9%

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

   13. Applied rewrites 76.9%

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

   if 1.80000000000000004 < phi2

   1. Initial program 98.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 78.4%

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 70.1%

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

   7. Applied rewrites 70.1%

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

Alternative 10: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0, \cos \phi_1\right)} + \lambda_1 \end{array} \]

FPCore

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

C

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	return Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * t_0), fma(cos(Float64(lambda2 - lambda1)), t_0, cos(phi1))) + lambda1)
end

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 76.2%

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

6. Applied rewrites 76.2%

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

7. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 76.9%

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

9. Applied rewrites 76.9%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 76.9%

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

    6. lift-pow.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 76.9%

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

11. Applied rewrites 76.9%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 76.9%

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

    6. lift-pow.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 76.9%

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

13. Applied rewrites 76.9%

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

Alternative 11: 68.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.4%

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 68.3%

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

7. Applied rewrites 68.3%

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

8. Taylor expanded in lambda1 around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-neg.f64 68.1%

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

10. Applied rewrites 68.1%

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

Alternative 12: 66.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.4%

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 68.3%

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

7. Applied rewrites 68.3%

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

8. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 66.0%

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

10. Applied rewrites 66.0%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 66.0%

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

12. Applied rewrites 66.0%

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

Alternative 13: 63.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) 2.0)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
}

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))), 2.0d0)
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))), 2.0);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), 2.0)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), 2.0))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.4%

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 68.3%

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

7. Applied rewrites 68.3%

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

8. Taylor expanded in lambda2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 62.5%

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

10. Applied rewrites 62.5%

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

11. Taylor expanded in lambda1 around 0

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

13. Applied rewrites 63.1%

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

Reproduce

herbie shell --seed 2025181 
(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)))))))