Midpoint on a great circle

Percentage Accurate: 98.6% → 99.6%

Time: 11.5s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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. cos-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 98.7

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

3. Applied rewrites 98.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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. cos-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 98.7

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

3. Applied rewrites 98.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-diff N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in lambda1 around 0

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

   1. lift-sin.f64 98.9

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

8. Applied rewrites 98.9%

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

Alternative 3: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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. cos-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 98.7

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

3. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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. cos-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 98.7

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

3. Applied rewrites 98.7%

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

4. Taylor expanded in lambda2 around 0

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

   1. lift-sin.f64 98.7

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

6. Applied rewrites 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(-\lambda_2\right), \cos \phi_2, \cos \phi_1\right)} + \lambda_1\\ \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 4 \cdot 10^{-162}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (atan2
           (* (sin (- lambda2)) (cos phi2))
           (fma (cos (- lambda2)) (cos phi2) (cos phi1)))
          lambda1)))
   (if (<= lambda2 -5e-54)
     t_0
     (if (<= lambda2 4e-162)
       (+
        lambda1
        (atan2
         (* (cos phi2) (sin (- lambda1 lambda2)))
         (+ (cos phi1) (* (cos phi2) 1.0))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((sin(-lambda2) * cos(phi2)), fma(cos(-lambda2), cos(phi2), cos(phi1))) + lambda1;
	double tmp;
	if (lambda2 <= -5e-54) {
		tmp = t_0;
	} else if (lambda2 <= 4e-162) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), fma(cos(Float64(-lambda2)), cos(phi2), cos(phi1))) + lambda1)
	tmp = 0.0
	if (lambda2 <= -5e-54)
		tmp = t_0;
	elseif (lambda2 <= 4e-162)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[lambda2, -5e-54], t$95$0, If[LessEqual[lambda2, 4e-162], 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] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.00000000000000015e-54 or 3.99999999999999982e-162 < lambda2

   1. Initial program 98.2%

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

   2. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 95.6

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

   9. Applied rewrites 95.6%

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

   if -5.00000000000000015e-54 < lambda2 < 3.99999999999999982e-162

   1. Initial program 99.5%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in lambda2 around 0

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

   7. Applied rewrites 98.0%

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

Alternative 7: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 97.9

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

4. Applied rewrites 97.9%

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

Alternative 8: 88.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 22.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \phi_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1
         (fma
          (-
           (*
            (fma (* phi2 phi2) -0.001388888888888889 0.041666666666666664)
            (* phi2 phi2))
           0.5)
          (* phi2 phi2)
          1.0))
        (t_2 (cos (- lambda1 lambda2))))
   (if (<= phi2 22.5)
     (+ lambda1 (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 t_2))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (* (cos phi2) t_2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma(((fma((phi2 * phi2), -0.001388888888888889, 0.041666666666666664) * (phi2 * phi2)) - 0.5), (phi2 * phi2), 1.0);
	double t_2 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 22.5) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * t_2)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (fma((phi1 * phi1), -0.5, 1.0) + (cos(phi2) * t_2)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(Float64(fma(Float64(phi2 * phi2), -0.001388888888888889, 0.041666666666666664) * Float64(phi2 * phi2)) - 0.5), Float64(phi2 * phi2), 1.0)
	t_2 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 22.5)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(cos(phi1) + Float64(t_1 * t_2))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(cos(phi2) * t_2))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 22.5], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 22.5

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\phi_2}^{2} + \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left({\phi_2}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-*.f64 83.3

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 83.3%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \left(\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}, {\color{blue}{\phi_2}}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\phi_2}^{2} + \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\left({\phi_2}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot {\phi_2}^{2} - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, {\phi_2}^{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      14. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(\phi_2 \cdot \phi_2\right) - \frac{1}{2}, \phi_2 \cdot \color{blue}{\phi_2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      15. lower-*.f64 83.4

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \color{blue}{\phi_2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. Applied rewrites 83.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 22.5 < phi2

   1. Initial program 98.4%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

Alternative 9: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 0.0062:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{\cos \phi_1 + t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \phi_2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 0.0062)
     (+ lambda1 (atan2 (* t_2 t_0) (+ (cos phi1) (* t_2 t_1))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (* (cos phi2) t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= 0.0062) {
		tmp = lambda1 + atan2((t_2 * t_0), (cos(phi1) + (t_2 * t_1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (fma((phi1 * phi1), -0.5, 1.0) + (cos(phi2) * t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= 0.0062)
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(cos(phi1) + Float64(t_2 * t_1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(cos(phi2) * t_1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 0.0062], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.00619999999999999978

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 0.00619999999999999978 < 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 phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

Alternative 10: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t\_1\\ \mathbf{if}\;\cos \phi_2 \leq -0.18:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos \phi_1 + \cos \phi_2 \cdot 1}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.99998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot t\_1}{\cos \phi_1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2) t_1)))
   (if (<= (cos phi2) -0.18)
     (+ lambda1 (atan2 t_2 (+ (cos phi1) (* (cos phi2) 1.0))))
     (if (<= (cos phi2) 0.99998)
       (+ lambda1 (atan2 t_2 (+ 1.0 (* (cos phi2) (cos lambda2)))))
       (+
        lambda1
        (atan2
         (* t_0 t_1)
         (+ (cos phi1) (* t_0 (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 t_2 = cos(phi2) * t_1;
	double tmp;
	if (cos(phi2) <= -0.18) {
		tmp = lambda1 + atan2(t_2, (cos(phi1) + (cos(phi2) * 1.0)));
	} else if (cos(phi2) <= 0.99998) {
		tmp = lambda1 + atan2(t_2, (1.0 + (cos(phi2) * cos(lambda2))));
	} else {
		tmp = lambda1 + atan2((t_0 * t_1), (cos(phi1) + (t_0 * 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))
	t_2 = Float64(cos(phi2) * t_1)
	tmp = 0.0
	if (cos(phi2) <= -0.18)
		tmp = Float64(lambda1 + atan(t_2, Float64(cos(phi1) + Float64(cos(phi2) * 1.0))));
	elseif (cos(phi2) <= 0.99998)
		tmp = Float64(lambda1 + atan(t_2, Float64(1.0 + Float64(cos(phi2) * cos(lambda2)))));
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_1), Float64(cos(phi1) + Float64(t_0 * 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]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.18], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.99998], N[(lambda1 + N[ArcTan[t$95$2 / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$1), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi2) < -0.17999999999999999

   1. Initial program 98.3%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in lambda2 around 0

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

   7. Applied rewrites 77.7%

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

   if -0.17999999999999999 < (cos.f64 phi2) < 0.99997999999999998

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. cos-neg-rev N/A

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

      2. lift-cos.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if 0.99997999999999998 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 11: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.9996:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.9996)
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (* (cos phi2) 1.0))))
     (+ lambda1 (atan2 t_0 (fma (cos (- lambda1 lambda2)) (cos phi2) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.9996) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * 1.0)));
	} else {
		tmp = lambda1 + atan2(t_0, fma(cos((lambda1 - lambda2)), cos(phi2), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.9996)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * 1.0))));
	else
		tmp = Float64(lambda1 + atan(t_0, fma(cos(Float64(lambda1 - lambda2)), cos(phi2), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9996], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99960000000000004

   1. Initial program 98.7%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in lambda2 around 0

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

   7. Applied rewrites 78.3%

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

   if 0.99960000000000004 < (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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

Alternative 12: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.99998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\cos \phi_1 + \cos \phi_2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot t\_1}{\cos \phi_1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

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 phi2) 0.99998)
     (+ lambda1 (atan2 (* (cos phi2) t_1) (+ (cos phi1) (* (cos phi2) 1.0))))
     (+
      lambda1
      (atan2 (* t_0 t_1) (+ (cos phi1) (* t_0 (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(phi2) <= 0.99998) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), (cos(phi1) + (cos(phi2) * 1.0)));
	} else {
		tmp = lambda1 + atan2((t_0 * t_1), (cos(phi1) + (t_0 * 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(phi2) <= 0.99998)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(cos(phi1) + Float64(cos(phi2) * 1.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_1), Float64(cos(phi1) + Float64(t_0 * 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[phi2], $MachinePrecision], 0.99998], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$1), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.99997999999999998

   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 lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in lambda2 around 0

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

   7. Applied rewrites 78.5%

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

   if 0.99997999999999998 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 13: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 3.2e+20)
     (+
      lambda1
      (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 (cos (- lambda1 lambda2))))))
     (+
      (atan2 (* t_0 (cos phi2)) (fma (cos lambda1) (cos phi2) 1.0))
      lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= 3.2e+20) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((t_0 * cos(phi2)), fma(cos(lambda1), cos(phi2), 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= 3.2e+20)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(cos(phi1) + Float64(t_1 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(atan(Float64(t_0 * cos(phi2)), fma(cos(lambda1), cos(phi2), 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 3.2e+20], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.2e20

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 3.2e20 < 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 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 77.7%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 67.8

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

   9. Applied rewrites 67.8%

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

Alternative 14: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \lambda_1 \cdot \lambda_1, 0.041666666666666664\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - 0.5, \lambda_1 \cdot \lambda_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 3.2e+20)
     (+
      lambda1
      (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+
        1.0
        (*
         (cos phi2)
         (fma
          (-
           (*
            (fma
             -0.001388888888888889
             (* lambda1 lambda1)
             0.041666666666666664)
            (* lambda1 lambda1))
           0.5)
          (* lambda1 lambda1)
          1.0))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= 3.2e+20) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + (cos(phi2) * fma(((fma(-0.001388888888888889, (lambda1 * lambda1), 0.041666666666666664) * (lambda1 * lambda1)) - 0.5), (lambda1 * lambda1), 1.0))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= 3.2e+20)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(cos(phi1) + Float64(t_1 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + Float64(cos(phi2) * fma(Float64(Float64(fma(-0.001388888888888889, Float64(lambda1 * lambda1), 0.041666666666666664) * Float64(lambda1 * lambda1)) - 0.5), Float64(lambda1 * lambda1), 1.0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 3.2e+20], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[(N[(-0.001388888888888889 * N[(lambda1 * lambda1), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(lambda1 * lambda1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(lambda1 * lambda1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.2e20

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 3.2e20 < 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 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 77.7%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \left(1 + \color{blue}{{\lambda_1}^{2} \cdot \left({\lambda_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) - \frac{1}{2}\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \left({\lambda_1}^{2} \cdot \left({\lambda_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \left(\left({\lambda_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) - \frac{1}{2}\right) \cdot {\lambda_1}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left({\lambda_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) - \frac{1}{2}, {\lambda_1}^{\color{blue}{2}}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left({\lambda_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\lambda_1}^{2}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\lambda_1}^{2} + \frac{1}{24}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {\lambda_1}^{2}, \frac{1}{24}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \lambda_1 \cdot \lambda_1, \frac{1}{24}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \lambda_1 \cdot \lambda_1, \frac{1}{24}\right) \cdot {\lambda_1}^{2} - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \lambda_1 \cdot \lambda_1, \frac{1}{24}\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \lambda_1 \cdot \lambda_1, \frac{1}{24}\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - \frac{1}{2}, {\lambda_1}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \lambda_1 \cdot \lambda_1, \frac{1}{24}\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - \frac{1}{2}, \lambda_1 \cdot \lambda_1, 1\right)} \]

      14. lower-*.f64 67.6

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \lambda_1 \cdot \lambda_1, 0.041666666666666664\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - 0.5, \lambda_1 \cdot \lambda_1, 1\right)} \]

   10. Applied rewrites 67.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \lambda_1 \cdot \lambda_1, 0.041666666666666664\right) \cdot \left(\lambda_1 \cdot \lambda_1\right) - 0.5, \color{blue}{\lambda_1 \cdot \lambda_1}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\lambda_1 \cdot \lambda_1, -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 3.2e+20)
     (+
      lambda1
      (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ 1.0 (* (cos phi2) (fma (* lambda1 lambda1) -0.5 1.0))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= 3.2e+20) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + (cos(phi2) * fma((lambda1 * lambda1), -0.5, 1.0))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= 3.2e+20)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(cos(phi1) + Float64(t_1 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + Float64(cos(phi2) * fma(Float64(lambda1 * lambda1), -0.5, 1.0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 3.2e+20], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * lambda1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.2e20

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 3.2e20 < 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 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 77.7%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

   8. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.6

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

   10. Applied rewrites 67.6%

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

Alternative 16: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.932:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \phi_2 \cdot \mathsf{fma}\left(\lambda_1 \cdot \lambda_1, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{1 \cdot t\_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.932)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ 1.0 (* (cos phi2) (fma (* lambda1 lambda1) -0.5 1.0)))))
     (+
      lambda1
      (atan2 (* 1.0 t_0) (+ (cos phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.932) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + (cos(phi2) * fma((lambda1 * lambda1), -0.5, 1.0))));
	} else {
		tmp = lambda1 + atan2((1.0 * t_0), (cos(phi1) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.932)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + Float64(cos(phi2) * fma(Float64(lambda1 * lambda1), -0.5, 1.0)))));
	else
		tmp = Float64(lambda1 + atan(Float64(1.0 * t_0), Float64(cos(phi1) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.932], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * lambda1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(1.0 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.932000000000000051

   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 77.3%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

   8. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.5

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

   10. Applied rewrites 67.5%

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

   if 0.932000000000000051 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in 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 78.9%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 76.9%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

   12. Applied rewrites 63.4%

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

   13. Taylor expanded in phi2 around 0

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

       1. pow2 N/A

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

       2. unpow2 N/A

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

       3. flip-+ N/A

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

       4. lower-+.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift-cos.f64 N/A

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

       7. lift--.f64 94.7

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

   15. Applied rewrites 94.7%

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

Alternative 17: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.945:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \phi_2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{1 \cdot t\_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.945)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ 1.0 (* (cos phi2) 1.0))))
     (+
      lambda1
      (atan2 (* 1.0 t_0) (+ (cos phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.945) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + (cos(phi2) * 1.0)));
	} else {
		tmp = lambda1 + atan2((1.0 * t_0), (cos(phi1) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.945d0) then
        tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0d0 + (cos(phi2) * 1.0d0)))
    else
        tmp = lambda1 + atan2((1.0d0 * t_0), (cos(phi1) + cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.945) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (1.0 + (Math.cos(phi2) * 1.0)));
	} else {
		tmp = lambda1 + Math.atan2((1.0 * t_0), (Math.cos(phi1) + Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.945:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (1.0 + (math.cos(phi2) * 1.0)))
	else:
		tmp = lambda1 + math.atan2((1.0 * t_0), (math.cos(phi1) + math.cos((lambda1 - lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.945)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + Float64(cos(phi2) * 1.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(1.0 * t_0), Float64(cos(phi1) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.945)
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + (cos(phi2) * 1.0)));
	else
		tmp = lambda1 + atan2((1.0 * t_0), (cos(phi1) + cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.945], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(1.0 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.944999999999999951

   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 77.2%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in lambda1 around 0

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

   10. Applied rewrites 67.5%

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

   if 0.944999999999999951 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in 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 79.0%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 77.1%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

   12. Applied rewrites 63.6%

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

   13. Taylor expanded in phi2 around 0

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

       1. pow2 N/A

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

       2. unpow2 N/A

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

       3. flip-+ N/A

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

       4. lower-+.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift-cos.f64 N/A

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

       7. lift--.f64 95.1

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

   15. Applied rewrites 95.1%

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

Alternative 18: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{1 + t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{1 \cdot t\_0}{\cos \phi_1 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= (cos phi2) -0.1)
     (+ lambda1 (atan2 (* t_2 t_0) (+ 1.0 (* t_2 t_1))))
     (+ lambda1 (atan2 (* 1.0 t_0) (+ (cos phi1) t_1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = lambda1 + atan2((t_2 * t_0), (1.0 + (t_2 * t_1)));
	} else {
		tmp = lambda1 + atan2((1.0 * t_0), (cos(phi1) + t_1));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(1.0 + Float64(t_2 * t_1))));
	else
		tmp = Float64(lambda1 + atan(Float64(1.0 * t_0), Float64(cos(phi1) + t_1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(1.0 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 98.3%

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

   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 76.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.8

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

   10. Applied rewrites 59.8%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in 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 78.7%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 71.7%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

   12. Applied rewrites 60.9%

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

   13. Taylor expanded in phi2 around 0

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

       1. pow2 N/A

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

       2. unpow2 N/A

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

       3. flip-+ N/A

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

       4. lower-+.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift-cos.f64 N/A

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

       7. lift--.f64 84.8

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

   15. Applied rewrites 84.8%

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

Alternative 19: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9991:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{1 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + 1 \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi1) 0.9991)
   (+
    lambda1
    (atan2 (* 1.0 (sin (- lambda2))) (+ (cos phi1) (* 1.0 (cos (- lambda2))))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ 1.0 (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi1) <= 0.9991) {
		tmp = lambda1 + atan2((1.0 * sin(-lambda2)), (cos(phi1) + (1.0 * cos(-lambda2))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0 + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (cos(phi1) <= 0.9991d0) then
        tmp = lambda1 + atan2((1.0d0 * sin(-lambda2)), (cos(phi1) + (1.0d0 * cos(-lambda2))))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0d0 + cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos(phi1) <= 0.9991) {
		tmp = lambda1 + Math.atan2((1.0 * Math.sin(-lambda2)), (Math.cos(phi1) + (1.0 * Math.cos(-lambda2))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (1.0 + Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi1) <= 0.9991:
		tmp = lambda1 + math.atan2((1.0 * math.sin(-lambda2)), (math.cos(phi1) + (1.0 * math.cos(-lambda2))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (1.0 + math.cos((lambda1 - lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi1) <= 0.9991)
		tmp = Float64(lambda1 + atan(Float64(1.0 * sin(Float64(-lambda2))), Float64(cos(phi1) + Float64(1.0 * cos(Float64(-lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(1.0 + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi1) <= 0.9991)
		tmp = lambda1 + atan2((1.0 * sin(-lambda2)), (cos(phi1) + (1.0 * cos(-lambda2))));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0 + cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9991], N[(lambda1 + N[ArcTan[N[(1.0 * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(1.0 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.999099999999999988

   1. Initial program 98.7%

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

   2. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 73.4%

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

   11. Taylor expanded in phi2 around 0

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

   13. Applied rewrites 71.3%

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

   if 0.999099999999999988 < (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 97.8%

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

   5. Taylor expanded in phi2 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. lift-cos.f64 N/A

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

      2. lift--.f64 78.2

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

   7. Applied rewrites 78.2%

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

Alternative 20: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{1 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + 1 \cdot \cos \left(-\lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= (cos phi2) -0.1)
     (+
      lambda1
      (atan2
       (* t_0 (sin (- lambda1 lambda2)))
       (+ 1.0 (* t_0 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* 1.0 (sin (- lambda2)))
       (+ (cos phi1) (* 1.0 (cos (- lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = lambda1 + atan2((t_0 * sin((lambda1 - lambda2))), (1.0 + (t_0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((1.0 * sin(-lambda2)), (cos(phi1) + (1.0 * cos(-lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(lambda1 + atan(Float64(t_0 * sin(Float64(lambda1 - lambda2))), Float64(1.0 + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(1.0 * sin(Float64(-lambda2))), Float64(cos(phi1) + Float64(1.0 * cos(Float64(-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]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(lambda1 + N[ArcTan[N[(t$95$0 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(1.0 * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(1.0 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 98.3%

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

   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 76.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.8

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

   10. Applied rewrites 59.8%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 78.6%

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

   11. Taylor expanded in phi2 around 0

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

   13. Applied rewrites 78.0%

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

Alternative 21: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{1 + t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{\mathsf{fma}\left(t\_1, 1, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= (cos phi2) -0.1)
     (+ lambda1 (atan2 (* t_2 t_0) (+ 1.0 (* t_2 t_1))))
     (+ (atan2 (* t_0 1.0) (fma t_1 1.0 1.0)) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = lambda1 + atan2((t_2 * t_0), (1.0 + (t_2 * t_1)));
	} else {
		tmp = atan2((t_0 * 1.0), fma(t_1, 1.0, 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(1.0 + Float64(t_2 * t_1))));
	else
		tmp = Float64(atan(Float64(t_0 * 1.0), fma(t_1, 1.0, 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / N[(t$95$1 * 1.0 + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 98.3%

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

   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 76.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.8

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

   10. Applied rewrites 59.8%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in 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 78.7%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 71.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 71.7

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

   12. Applied rewrites 71.7%

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

Alternative 22: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{1 + t\_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot 1}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), 1, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= (cos phi2) -0.1)
     (+ lambda1 (atan2 (* t_1 t_0) (+ 1.0 (* t_1 (cos lambda1)))))
     (+ (atan2 (* t_0 1.0) (fma (cos (- lambda1 lambda2)) 1.0 1.0)) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = lambda1 + atan2((t_1 * t_0), (1.0 + (t_1 * cos(lambda1))));
	} else {
		tmp = atan2((t_0 * 1.0), fma(cos((lambda1 - lambda2)), 1.0, 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(1.0 + Float64(t_1 * cos(lambda1)))));
	else
		tmp = Float64(atan(Float64(t_0 * 1.0), fma(cos(Float64(lambda1 - lambda2)), 1.0, 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(t$95$0 * 1.0), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 98.3%

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

   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 76.6%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lift-cos.f64 67.0

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

   7. Applied rewrites 67.0%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 55.8

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

   10. Applied rewrites 55.8%

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

   11. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 59.7

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

   13. Applied rewrites 59.7%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.7%

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

   2. Taylor expanded in 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 78.7%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 71.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 71.7

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

   12. Applied rewrites 71.7%

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

Alternative 23: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in 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 78.2%

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

5. Taylor expanded in phi2 around 0

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

7. Applied rewrites 66.5%

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

8. Taylor expanded in phi2 around 0

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

10. Applied rewrites 66.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 66.5

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

12. Applied rewrites 66.5%

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

Alternative 24: 61.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in 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 78.2%

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

5. Taylor expanded in phi2 around 0

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

7. Applied rewrites 66.5%

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

8. Taylor expanded in phi2 around 0

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

10. Applied rewrites 66.5%

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

11. Taylor expanded in lambda1 around inf

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

13. Applied rewrites 54.8%

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

14. Taylor expanded in lambda1 around inf

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{1 \cdot \sin \lambda_1}{1 + 1 \cdot \cos \color{blue}{\lambda_1}} \]
15. Step-by-step derivation

16. Applied rewrites 54.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{1 \cdot \sin \lambda_1}{1 + 1 \cdot \cos \color{blue}{\lambda_1}} \]

17. Taylor expanded in phi2 around 0

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

    1. lift-sin.f64 N/A

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

    2. lift--.f64 61.4

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

19. Applied rewrites 61.4%

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

Alternative 25: 52.2% accurate, 192.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2) :precision binary64 lambda1)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1

Julia

function code(lambda1, lambda2, phi1, phi2)
	return lambda1
end

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := lambda1

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in lambda1 around inf

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

4. Applied rewrites 52.2%

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

Reproduce

herbie shell --seed 2025123 
(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)))))))