Midpoint on a great circle

Percentage Accurate: 98.7% → 98.7%

Time: 10.0s

Alternatives: 16

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: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% 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 98.7%

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

Alternative 2: 98.0% 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 98.7%

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

2. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 3: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -1000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\left(\lambda_2 \cdot \lambda_1\right) \cdot -0.5 - 1, \lambda_2, \lambda_1\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          lambda1
          (atan2
           (* (cos phi2) (sin (- lambda2)))
           (+ (cos phi1) (* (cos phi2) (cos lambda2)))))))
   (if (<= lambda2 -1000000000000.0)
     t_0
     (if (<= lambda2 5e-97)
       (+
        lambda1
        (atan2
         (*
          (cos phi2)
          (fma (- (* (* lambda2 lambda1) -0.5) 1.0) lambda2 lambda1))
         (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	double tmp;
	if (lambda2 <= -1000000000000.0) {
		tmp = t_0;
	} else if (lambda2 <= 5e-97) {
		tmp = lambda1 + atan2((cos(phi2) * fma((((lambda2 * lambda1) * -0.5) - 1.0), lambda2, lambda1)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))))
	tmp = 0.0
	if (lambda2 <= -1000000000000.0)
		tmp = t_0;
	elseif (lambda2 <= 5e-97)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * fma(Float64(Float64(Float64(lambda2 * lambda1) * -0.5) - 1.0), lambda2, lambda1)), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1000000000000.0], t$95$0, If[LessEqual[lambda2, 5e-97], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[(N[(lambda2 * lambda1), $MachinePrecision] * -0.5), $MachinePrecision] - 1.0), $MachinePrecision] * lambda2 + lambda1), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1e12 or 4.9999999999999995e-97 < lambda2

   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 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 lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   10. Applied rewrites 71.9%

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

   11. Taylor expanded in lambda1 around 0

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

       1. lower-+.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. cos-neg N/A

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

       6. lower-cos.f64 87.9

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

   13. Applied rewrites 87.9%

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

   if -1e12 < lambda2 < 4.9999999999999995e-97

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.7

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

   7. Applied rewrites 75.7%

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

Alternative 4: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{t\_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) + \cos \phi_2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi2 5.5e-8)
     (+
      lambda1
      (atan2 (* (fma (* phi2 phi2) -0.5 1.0) t_0) (+ t_1 (cos phi1))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+
        (fma
         (-
          (*
           (fma -0.001388888888888889 (* phi1 phi1) 0.041666666666666664)
           (* phi1 phi1))
          0.5)
         (* phi1 phi1)
         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 tmp;
	if (phi2 <= 5.5e-8) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (t_1 + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (fma(((fma(-0.001388888888888889, (phi1 * phi1), 0.041666666666666664) * (phi1 * phi1)) - 0.5), (phi1 * phi1), 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))
	tmp = 0.0
	if (phi2 <= 5.5e-8)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(t_1 + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(phi1 * phi1), 0.041666666666666664) * Float64(phi1 * phi1)) - 0.5), Float64(phi1 * phi1), 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]}, If[LessEqual[phi2, 5.5e-8], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(-0.001388888888888889 * N[(phi1 * phi1), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 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 < 5.5000000000000003e-8

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 74.7

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

   10. Applied rewrites 74.7%

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

   if 5.5000000000000003e-8 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right)} + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{t\_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \phi_2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi2 5.5e-8)
     (+
      lambda1
      (atan2 (* (fma (* phi2 phi2) -0.5 1.0) t_0) (+ t_1 (cos phi1))))
     (+
      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 tmp;
	if (phi2 <= 5.5e-8) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (t_1 + cos(phi1)));
	} 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))
	tmp = 0.0
	if (phi2 <= 5.5e-8)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(t_1 + cos(phi1))));
	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]}, If[LessEqual[phi2, 5.5e-8], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 + N[Cos[phi1], $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 < 5.5000000000000003e-8

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 74.7

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

   10. Applied rewrites 74.7%

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

   if 5.5000000000000003e-8 < 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

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)} + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   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 79.3

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

   4. Applied rewrites 79.3%

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

Alternative 6: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 0.0146:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 + \cos \phi_2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 0.0146)
     (+ lambda1 (atan2 t_1 (+ 1.0 (* (cos phi2) t_0))))
     (+ lambda1 (atan2 t_1 (+ t_0 (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.0146) {
		tmp = lambda1 + atan2(t_1, (1.0 + (cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= 0.0146d0) then
        tmp = lambda1 + atan2(t_1, (1.0d0 + (cos(phi2) * t_0)))
    else
        tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.0146) {
		tmp = lambda1 + Math.atan2(t_1, (1.0 + (Math.cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (t_0 + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 0.0146:
		tmp = lambda1 + math.atan2(t_1, (1.0 + (math.cos(phi2) * t_0)))
	else:
		tmp = lambda1 + math.atan2(t_1, (t_0 + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 0.0146)
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 + Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= 0.0146)
		tmp = lambda1 + atan2(t_1, (1.0 + (cos(phi2) * t_0)));
	else
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 0.0146], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 0.0146000000000000001

   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 77.7%

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

   if 0.0146000000000000001 < phi1

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 7: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 0.0146:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 0.0146)
     (+ lambda1 (atan2 t_1 (fma t_0 (cos phi2) 1.0)))
     (+ lambda1 (atan2 t_1 (+ t_0 (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.0146) {
		tmp = lambda1 + atan2(t_1, fma(t_0, cos(phi2), 1.0));
	} else {
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 0.0146)
		tmp = Float64(lambda1 + atan(t_1, fma(t_0, cos(phi2), 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 0.0146], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 0.0146000000000000001

   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

      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 77.7

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

   4. Applied rewrites 77.7%

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

   if 0.0146000000000000001 < phi1

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 8: 80.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.1 \cdot 10^{+50}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{1 + \cos \phi_2 \cdot \cos \left(-\lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.1e+50)
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ (cos (- lambda1 lambda2)) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda2)))
     (+ 1.0 (* (cos phi2) (cos (- lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.1e+50) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (1.0 + (cos(phi2) * cos(-lambda2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3.1d+50) then
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (1.0d0 + (cos(phi2) * cos(-lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.1e+50) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos((lambda1 - lambda2)) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (1.0 + (Math.cos(phi2) * Math.cos(-lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.1e+50:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos((lambda1 - lambda2)) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), (1.0 + (math.cos(phi2) * math.cos(-lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.1e+50)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(Float64(lambda1 - lambda2)) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(1.0 + Float64(cos(phi2) * cos(Float64(-lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.1e+50)
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (1.0 + (cos(phi2) * cos(-lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.1e+50], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.10000000000000003e50

   1. Initial program 98.7%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if 3.10000000000000003e50 < 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 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 lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   10. Applied rewrites 71.9%

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

Alternative 9: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 46000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{1 + \cos \phi_2 \cdot \cos \left(-\lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 46000000000000.0)
     (+
      lambda1
      (atan2
       (* t_0 (sin (- lambda1 lambda2)))
       (+ (cos phi1) (* t_0 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* (cos phi2) (sin (- lambda2)))
       (+ 1.0 (* (cos phi2) (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 (phi2 <= 46000000000000.0) {
		tmp = lambda1 + atan2((t_0 * sin((lambda1 - lambda2))), (cos(phi1) + (t_0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (1.0 + (cos(phi2) * 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 (phi2 <= 46000000000000.0)
		tmp = Float64(lambda1 + atan(Float64(t_0 * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(1.0 + Float64(cos(phi2) * 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[phi2, 46000000000000.0], N[(lambda1 + N[ArcTan[N[(t$95$0 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.6e13

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   if 4.6e13 < 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 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 lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   10. Applied rewrites 71.9%

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

Alternative 10: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 46000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi2 46000000000000.0)
     (+
      lambda1
      (atan2 (* t_1 (sin (- lambda1 lambda2))) (+ (cos phi1) (* t_1 t_0))))
     (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda2))) (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi2 <= 46000000000000.0) {
		tmp = lambda1 + atan2((t_1 * sin((lambda1 - lambda2))), (cos(phi1) + (t_1 * t_0)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi2 <= 46000000000000.0)
		tmp = Float64(lambda1 + atan(Float64(t_1 * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(t_1 * t_0))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi1) + t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 46000000000000.0], N[(lambda1 + N[ArcTan[N[(t$95$1 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.6e13

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   if 4.6e13 < 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 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 lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   10. Applied rewrites 71.9%

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

   11. Taylor expanded in phi2 around 0

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

       1. lower-+.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift--.f64 72.0

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

   13. Applied rewrites 72.0%

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

Alternative 11: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_2 \leq 46000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \sin \left(\left(\lambda_1 - -0.5 \cdot \pi\right) - \lambda_2\right)}\\ \end{array} \end{array} \]

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 46000000000000.0)
     (+
      lambda1
      (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ 1.0 (sin (- (- lambda1 (* -0.5 PI)) lambda2))))))))

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 <= 46000000000000.0) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + sin(((lambda1 - (-0.5 * ((double) M_PI))) - lambda2))));
	}
	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 <= 46000000000000.0)
		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 + sin(Float64(Float64(lambda1 - Float64(-0.5 * pi)) - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 46000000000000.0], 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[Sin[N[(N[(lambda1 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.6e13

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   if 4.6e13 < 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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. sin-cos-mult N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 62.5

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

   9. Applied rewrites 62.5%

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

Alternative 12: 76.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 46000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \sin \left(\left(\lambda_1 - -0.5 \cdot \pi\right) - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 46000000000000.0)
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (+ (cos (- lambda1 lambda2)) (cos phi1))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ 1.0 (sin (- (- lambda1 (* -0.5 PI)) lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 46000000000000.0) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (cos((lambda1 - lambda2)) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + sin(((lambda1 - (-0.5 * ((double) M_PI))) - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 46000000000000.0)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(cos(Float64(lambda1 - lambda2)) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + sin(Float64(Float64(lambda1 - Float64(-0.5 * pi)) - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 46000000000000.0], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Sin[N[(N[(lambda1 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.6e13

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 74.7

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

   10. Applied rewrites 74.7%

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

   if 4.6e13 < 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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. sin-cos-mult N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 62.5

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

   9. Applied rewrites 62.5%

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

Alternative 13: 67.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.994:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{1 + \sin \left(\left(\lambda_1 - -0.5 \cdot \pi\right) - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot t\_1}{1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0))
        (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.994)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (+ 1.0 (sin (- (- lambda1 (* -0.5 PI)) lambda2)))))
     (+
      lambda1
      (atan2 (* t_0 t_1) (+ 1.0 (* t_0 (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.994) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), (1.0 + sin(((lambda1 - (-0.5 * ((double) M_PI))) - lambda2))));
	} else {
		tmp = lambda1 + atan2((t_0 * t_1), (1.0 + (t_0 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.994)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(1.0 + sin(Float64(Float64(lambda1 - Float64(-0.5 * pi)) - lambda2)))));
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_1), Float64(1.0 + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.994], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(1.0 + N[Sin[N[(N[(lambda1 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$1), $MachinePrecision] / N[(1.0 + 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) < 0.99399999999999999

   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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. sin-cos-mult N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 62.5

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

   9. Applied rewrites 62.5%

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

   if 0.99399999999999999 < (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 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 phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 64.9

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 14: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_2 := \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.15:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{1 + t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (fma (* phi2 phi2) -0.5 1.0))
        (t_2
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0)))
   (if (<= (cos phi2) -0.15)
     (+
      lambda1
      (atan2
       (* t_1 t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (* t_1 (cos lambda1)))))
     (+
      lambda1
      (atan2 (* t_2 t_0) (+ 1.0 (* t_2 (cos (- lambda1 lambda2)))))))))

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 t_2 = fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0);
	double tmp;
	if (cos(phi2) <= -0.15) {
		tmp = lambda1 + atan2((t_1 * t_0), (fma((phi1 * phi1), -0.5, 1.0) + (t_1 * cos(lambda1))));
	} else {
		tmp = lambda1 + atan2((t_2 * t_0), (1.0 + (t_2 * cos((lambda1 - lambda2)))));
	}
	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)
	t_2 = fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	tmp = 0.0
	if (cos(phi2) <= -0.15)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + Float64(t_1 * cos(lambda1)))));
	else
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(1.0 + Float64(t_2 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.15], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.149999999999999994

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   11. Taylor expanded in lambda2 around 0

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

       1. lower-cos.f64 58.4

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

   13. Applied rewrites 58.4%

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

   if -0.149999999999999994 < (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 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 phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 64.9

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 15: 64.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;\phi_1 \leq 0.0145:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot t\_1}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), t\_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot \sin \left(-\lambda_2\right)}{1 + t\_0 \cdot \cos \left(-\lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* phi2 phi2)))) (t_1 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= phi1 0.0145)
     (+
      (atan2
       (* (sin (- lambda1 lambda2)) t_1)
       (fma (cos (- lambda1 lambda2)) t_1 (fma (* phi1 phi1) -0.5 1.0)))
      lambda1)
     (+
      lambda1
      (atan2 (* t_0 (sin (- lambda2))) (+ 1.0 (* t_0 (cos (- lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - (0.5 * (phi2 * phi2));
	double t_1 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (phi1 <= 0.0145) {
		tmp = atan2((sin((lambda1 - lambda2)) * t_1), fma(cos((lambda1 - lambda2)), t_1, fma((phi1 * phi1), -0.5, 1.0))) + lambda1;
	} else {
		tmp = lambda1 + atan2((t_0 * sin(-lambda2)), (1.0 + (t_0 * cos(-lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(phi2 * phi2)))
	t_1 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (phi1 <= 0.0145)
		tmp = Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * t_1), fma(cos(Float64(lambda1 - lambda2)), t_1, fma(Float64(phi1 * phi1), -0.5, 1.0))) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * sin(Float64(-lambda2))), Float64(1.0 + Float64(t_0 * cos(Float64(-lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[phi1, 0.0145], N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * t$95$1 + N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 0.0145000000000000007

   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 75.2

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

   4. Applied rewrites 75.2%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 63.2

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

   12. Applied rewrites 63.2%

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

   if 0.0145000000000000007 < phi1

   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 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 lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   10. Applied rewrites 71.9%

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

   11. Taylor expanded in phi2 around 0

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

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

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

       10. lower--.f64 N/A

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

       11. metadata-eval N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 62.1

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

   13. Applied rewrites 62.1%

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

   14. Taylor expanded in phi2 around 0

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

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

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

       10. lower--.f64 N/A

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

       11. metadata-eval N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 62.9

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

   16. Applied rewrites 62.9%

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

Alternative 16: 62.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = 1.0 - (0.5 * (phi2 * phi2))
	return lambda1 + math.atan2((t_0 * math.sin(-lambda2)), (1.0 + (t_0 * math.cos(-lambda2))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	t_0 = 1.0 - (0.5 * (phi2 * phi2));
	tmp = lambda1 + atan2((t_0 * sin(-lambda2)), (1.0 + (t_0 * cos(-lambda2))));
end

Wolfram

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

Derivation

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 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 lambda1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 71.9

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

7. Applied rewrites 71.9%

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

8. Taylor expanded in lambda1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 71.9

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

10. Applied rewrites 71.9%

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

11. Taylor expanded in phi2 around 0

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

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

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

    2. lift-/.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. sin-sum-rev N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-/.f64 N/A

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

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

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

    10. lower--.f64 N/A

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

    11. metadata-eval N/A

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

    12. lower-*.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 62.1

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

13. Applied rewrites 62.1%

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

14. Taylor expanded in phi2 around 0

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

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

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

    2. lift-/.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. sin-sum-rev N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-/.f64 N/A

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

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

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

    10. lower--.f64 N/A

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

    11. metadata-eval N/A

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

    12. lower-*.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 62.9

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

16. Applied rewrites 62.9%

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

Reproduce

herbie shell --seed 2025143 
(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)))))))