Midpoint on a great circle

Percentage Accurate: 98.7% → 98.7%

Time: 15.6s

Alternatives: 20

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 99.1%

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

Alternative 2: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \leq -3.14:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 + \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* (cos phi2) t_0)))
   (if (<=
        (+
         lambda1
         (atan2 t_1 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
        -3.14)
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda2 lambda1)))))
     (+ lambda1 (atan2 t_1 (+ 1.0 (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi2) * t_0;
	double tmp;
	if ((lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))) <= -3.14) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + atan2(t_1, (1.0 + cos(lambda2)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	tmp = 0.0
	if (Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))) <= -3.14)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 + cos(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[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -3.14], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 89.3

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

   11. Applied rewrites 89.3%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 94.6%

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

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

   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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 62.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \leq -3.14:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_0}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.665:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(1, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) -0.4)
     (+ lambda1 (atan2 t_1 (+ (fma (* phi1 phi1) -0.5 1.0) t_0)))
     (if (<= (cos phi2) 0.665)
       (+
        lambda1
        (atan2
         t_1
         (fma
          (* (- (* 0.041666666666666664 (* phi1 phi1)) 0.5) phi1)
          phi1
          (fma 1.0 (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 (cos(phi2) <= -0.4) {
		tmp = lambda1 + atan2(t_1, (fma((phi1 * phi1), -0.5, 1.0) + t_0));
	} else if (cos(phi2) <= 0.665) {
		tmp = lambda1 + atan2(t_1, fma((((0.041666666666666664 * (phi1 * phi1)) - 0.5) * phi1), phi1, fma(1.0, 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 (cos(phi2) <= -0.4)
		tmp = Float64(lambda1 + atan(t_1, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + t_0)));
	elseif (cos(phi2) <= 0.665)
		tmp = Float64(lambda1 + atan(t_1, fma(Float64(Float64(Float64(0.041666666666666664 * Float64(phi1 * phi1)) - 0.5) * phi1), phi1, fma(1.0, 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[N[Cos[phi2], $MachinePrecision], -0.4], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.665], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(N[(0.041666666666666664 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * phi1), $MachinePrecision] * phi1 + N[(1.0 * N[Cos[phi2], $MachinePrecision] + 1.0), $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 3 regimes

2. if (cos.f64 phi2) < -0.40000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 68.4%

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

   if -0.40000000000000002 < (cos.f64 phi2) < 0.66500000000000004

   1. Initial program 96.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in lambda2 around 0

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

   11. Applied rewrites 80.6%

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

   if 0.66500000000000004 < (cos.f64 phi2)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 91.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.665:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(1, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.999999995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.041666666666666664, -0.5\right), \phi_2 \cdot \phi_2, t\_1 + \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.999999995)
     (+
      lambda1
      (atan2 (* (cos phi2) t_0) (fma (cos lambda1) (cos phi2) (cos phi1))))
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (fma
        (* t_1 (fma (* phi2 phi2) 0.041666666666666664 -0.5))
        (* phi2 phi2)
        (+ t_1 (cos phi1))))))))

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 (cos(phi2) <= 0.999999995) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), fma(cos(lambda1), cos(phi2), cos(phi1)));
	} else {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), fma((t_1 * fma((phi2 * phi2), 0.041666666666666664, -0.5)), (phi2 * phi2), (t_1 + cos(phi1))));
	}
	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 (cos(phi2) <= 0.999999995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), fma(cos(lambda1), cos(phi2), cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), fma(Float64(t_1 * fma(Float64(phi2 * phi2), 0.041666666666666664, -0.5)), Float64(phi2 * phi2), Float64(t_1 + cos(phi1)))));
	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[N[Cos[phi2], $MachinePrecision], 0.999999995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(t$95$1 * N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + N[(t$95$1 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.99999999500000003

   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

   3. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 0.99999999500000003 < (cos.f64 phi2)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.999999995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.041666666666666664, -0.5\right), \phi_2 \cdot \phi_2, \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, \mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), t\_0, \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.9995)
     (+
      lambda1
      (atan2
       t_1
       (fma
        (-
         (*
          (fma -0.001388888888888889 (* phi1 phi1) 0.041666666666666664)
          (* phi1 phi1))
         0.5)
        (* phi1 phi1)
        (fma t_0 (cos phi2) 1.0))))
     (+
      lambda1
      (atan2 t_1 (fma (fma (* phi2 phi2) -0.5 1.0) 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 (cos(phi2) <= 0.9995) {
		tmp = lambda1 + atan2(t_1, fma(((fma(-0.001388888888888889, (phi1 * phi1), 0.041666666666666664) * (phi1 * phi1)) - 0.5), (phi1 * phi1), fma(t_0, cos(phi2), 1.0)));
	} else {
		tmp = lambda1 + atan2(t_1, fma(fma((phi2 * phi2), -0.5, 1.0), 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 (cos(phi2) <= 0.9995)
		tmp = Float64(lambda1 + atan(t_1, fma(Float64(Float64(fma(-0.001388888888888889, Float64(phi1 * phi1), 0.041666666666666664) * Float64(phi1 * phi1)) - 0.5), Float64(phi1 * phi1), fma(t_0, cos(phi2), 1.0))));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(fma(Float64(phi2 * phi2), -0.5, 1.0), 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[N[Cos[phi2], $MachinePrecision], 0.9995], N[(lambda1 + N[ArcTan[t$95$1 / 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] + N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.99950000000000006

   1. Initial program 98.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. sin-sum N/A

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

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.7%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\frac{{\left(\sin \phi_1 \cdot 0\right)}^{3} + {\left(\cos \phi_1 \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \phi_1 \cdot 0, \sin \phi_1 \cdot 0, \left(\cos \phi_1 \cdot 1\right) \cdot \left(\cos \phi_1 \cdot 1\right) - \left(\sin \phi_1 \cdot 0\right) \cdot \left(\cos \phi_1 \cdot 1\right)\right)}} + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 80.7%

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

   if 0.99950000000000006 < (cos.f64 phi2)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, \mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t\_1\\ \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_0}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.967:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(1, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2) t_1)))
   (if (<= (cos phi2) -0.4)
     (+ lambda1 (atan2 t_2 (+ (fma (* phi1 phi1) -0.5 1.0) t_0)))
     (if (<= (cos phi2) 0.967)
       (+
        lambda1
        (atan2
         t_2
         (fma
          (* (- (* 0.041666666666666664 (* phi1 phi1)) 0.5) phi1)
          phi1
          (fma 1.0 (cos phi2) 1.0))))
       (+
        lambda1
        (atan2
         (*
          (fma
           (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
           (* phi2 phi2)
           1.0)
          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 = sin((lambda1 - lambda2));
	double t_2 = cos(phi2) * t_1;
	double tmp;
	if (cos(phi2) <= -0.4) {
		tmp = lambda1 + atan2(t_2, (fma((phi1 * phi1), -0.5, 1.0) + t_0));
	} else if (cos(phi2) <= 0.967) {
		tmp = lambda1 + atan2(t_2, fma((((0.041666666666666664 * (phi1 * phi1)) - 0.5) * phi1), phi1, fma(1.0, cos(phi2), 1.0)));
	} else {
		tmp = lambda1 + atan2((fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0) * t_1), (t_0 + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * t_1)
	tmp = 0.0
	if (cos(phi2) <= -0.4)
		tmp = Float64(lambda1 + atan(t_2, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + t_0)));
	elseif (cos(phi2) <= 0.967)
		tmp = Float64(lambda1 + atan(t_2, fma(Float64(Float64(Float64(0.041666666666666664 * Float64(phi1 * phi1)) - 0.5) * phi1), phi1, fma(1.0, cos(phi2), 1.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0) * 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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.4], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.967], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[(N[(N[(0.041666666666666664 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * phi1), $MachinePrecision] * phi1 + N[(1.0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi2) < -0.40000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 68.4%

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

   if -0.40000000000000002 < (cos.f64 phi2) < 0.966999999999999971

   1. Initial program 97.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 76.0%

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

   9. Taylor expanded in lambda2 around 0

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

   11. Applied rewrites 70.3%

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

   if 0.966999999999999971 < (cos.f64 phi2)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.967:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(1, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0\\ \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (fma (* phi2 phi2) -0.5 1.0) t_0)))
   (if (<= (cos phi2) -0.4)
     (+
      lambda1
      (atan2 t_1 (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda2 lambda1)))))
     (if (<= (cos phi2) 0.9995)
       (+ lambda1 (atan2 (* (cos phi2) t_0) (+ 1.0 (cos lambda2))))
       (+ lambda1 (atan2 t_1 (+ (cos (- lambda1 lambda2)) (cos phi1))))))))

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) * t_0;
	double tmp;
	if (cos(phi2) <= -0.4) {
		tmp = lambda1 + atan2(t_1, (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda2 - lambda1))));
	} else if (cos(phi2) <= 0.9995) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + cos(lambda2)));
	} else {
		tmp = lambda1 + atan2(t_1, (cos((lambda1 - lambda2)) + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0)
	tmp = 0.0
	if (cos(phi2) <= -0.4)
		tmp = Float64(lambda1 + atan(t_1, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda2 - lambda1)))));
	elseif (cos(phi2) <= 0.9995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + cos(lambda2))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(Float64(lambda1 - lambda2)) + cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.4], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi2) < -0.40000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 55.2

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

   11. Applied rewrites 55.2%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 62.9%

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

   if -0.40000000000000002 < (cos.f64 phi2) < 0.99950000000000006

   1. Initial program 98.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 62.2%

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

   if 0.99950000000000006 < (cos.f64 phi2)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

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

   8. Applied rewrites 99.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0\\ \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (fma (* phi2 phi2) -0.5 1.0) t_0)))
   (if (<= (cos phi2) -0.4)
     (+
      lambda1
      (atan2 t_1 (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda2 lambda1)))))
     (if (<= (cos phi2) 0.9995)
       (+ lambda1 (atan2 (* (cos phi2) t_0) (+ 1.0 (cos lambda2))))
       (+ lambda1 (atan2 t_1 (+ (cos lambda2) (cos phi1))))))))

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) * t_0;
	double tmp;
	if (cos(phi2) <= -0.4) {
		tmp = lambda1 + atan2(t_1, (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda2 - lambda1))));
	} else if (cos(phi2) <= 0.9995) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + cos(lambda2)));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(lambda2) + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0)
	tmp = 0.0
	if (cos(phi2) <= -0.4)
		tmp = Float64(lambda1 + atan(t_1, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda2 - lambda1)))));
	elseif (cos(phi2) <= 0.9995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + cos(lambda2))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(lambda2) + cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.4], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi2) < -0.40000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 55.2

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

   11. Applied rewrites 55.2%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 62.9%

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

   if -0.40000000000000002 < (cos.f64 phi2) < 0.99950000000000006

   1. Initial program 98.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 62.2%

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

   if 0.99950000000000006 < (cos.f64 phi2)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 78.3

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

   11. Applied rewrites 78.3%

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

   12. Taylor expanded in lambda1 around 0

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

   14. Applied rewrites 98.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.9995)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (fma (* -0.5 phi1) phi1 (fma t_0 (cos phi2) 1.0))))
     (+
      lambda1
      (atan2
       (*
        (fma (- (* (* phi2 phi2) 0.041666666666666664) 0.5) (* phi2 phi2) 1.0)
        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 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.9995) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), fma((-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0)));
	} else {
		tmp = lambda1 + atan2((fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0) * t_1), (t_0 + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.9995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), fma(Float64(-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0) * 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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(-0.5 * phi1), $MachinePrecision] * phi1 + N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.99950000000000006

   1. Initial program 98.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   if 0.99950000000000006 < (cos.f64 phi2)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.1

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

   8. Applied rewrites 99.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.9995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 99.1%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Final simplification 98.1%

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

Alternative 11: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 0.0245:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), t\_0, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 0.0245)
     (+ lambda1 (atan2 t_1 (fma (fma (* phi2 phi2) -0.5 1.0) t_0 (cos phi1))))
     (+
      lambda1
      (atan2 t_1 (fma (* -0.5 phi1) phi1 (fma t_0 (cos phi2) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 0.0245) {
		tmp = lambda1 + atan2(t_1, fma(fma((phi2 * phi2), -0.5, 1.0), t_0, cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_1, fma((-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= 0.0245)
		tmp = Float64(lambda1 + atan(t_1, fma(fma(Float64(phi2 * phi2), -0.5, 1.0), t_0, cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(Float64(-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0))));
	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[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 0.0245], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(-0.5 * phi1), $MachinePrecision] * phi1 + N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.024500000000000001

   1. Initial program 98.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.6%

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

   if 0.024500000000000001 < phi2

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 82.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0245:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.9954:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.9954)
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (+ (cos lambda1) (cos phi1))))
     (+
      lambda1
      (atan2 (* (cos phi2) t_0) (+ 1.0 (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.9954) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (cos(lambda1) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.9954)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(cos(lambda1) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9954], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[lambda1], $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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.995399999999999952

   1. Initial program 98.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 51.1

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

   11. Applied rewrites 51.1%

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

   12. Taylor expanded in lambda2 around 0

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

   14. Applied rewrites 62.4%

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

   if 0.995399999999999952 < (cos.f64 phi1)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 83.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9954:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 0.11:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 0.11)
     (+ 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 <= 0.11) {
		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 <= 0.11)
		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, 0.11], 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 < 0.110000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   if 0.110000000000000001 < phi1

   1. Initial program 97.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 77.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 0.11:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) -0.4)
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda2 lambda1)))))
     (+
      lambda1
      (atan2 (* (cos phi2) t_0) (+ 1.0 (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= -0.4) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (1.0 + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= -0.4)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(1.0 + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.4], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.40000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 55.2

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

   11. Applied rewrites 55.2%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 62.9%

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

   if -0.40000000000000002 < (cos.f64 phi2)

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 72.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.21:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot t\_0}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) -0.21)
     (+
      lambda1
      (atan2
       (* (fma (* phi2 phi2) -0.5 1.0) t_0)
       (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda2 lambda1)))))
     (+
      lambda1
      (atan2
       (*
        (fma (- (* (* phi2 phi2) 0.041666666666666664) 0.5) (* phi2 phi2) 1.0)
        t_0)
       (+ 1.0 (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= -0.21) {
		tmp = lambda1 + atan2((fma((phi2 * phi2), -0.5, 1.0) * t_0), (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + atan2((fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0) * t_0), (1.0 + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= -0.21)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(phi2 * phi2), -0.5, 1.0) * t_0), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0) * t_0), Float64(1.0 + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.21], N[(lambda1 + N[ArcTan[N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.209999999999999992

   1. Initial program 98.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 56.2

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

   11. Applied rewrites 56.2%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 63.0%

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

   if -0.209999999999999992 < (cos.f64 phi2)

   1. Initial program 99.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 72.5%

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

   9. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 71.0

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

   11. Applied rewrites 71.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.21:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 17:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot t\_0}{t\_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi2 17.0)
     (+
      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) 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 <= 17.0) {
		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) + 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 <= 17.0)
		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) + 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, 17.0], 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] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 17

   1. Initial program 98.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

   if 17 < phi2

   1. Initial program 99.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 69.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 17:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 68.9%

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

9. Taylor expanded in phi2 around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 66.7

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

11. Applied rewrites 66.7%

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

12. Taylor expanded in phi1 around 0

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

14. Applied rewrites 66.8%

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

15. Final simplification 66.8%

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

Alternative 18: 65.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 68.9%

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

9. Taylor expanded in phi2 around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 66.7

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

11. Applied rewrites 66.7%

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

12. Final simplification 66.7%

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

Alternative 19: 65.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 68.9%

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

9. Taylor expanded in phi2 around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 66.7

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

11. Applied rewrites 66.7%

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

12. Taylor expanded in lambda1 around 0

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

14. Applied rewrites 66.3%

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

15. Final simplification 66.3%

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

Alternative 20: 55.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 68.9%

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

9. Taylor expanded in phi2 around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 66.7

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

11. Applied rewrites 66.7%

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

12. Taylor expanded in lambda2 around 0

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

    1. lower-sin.f64 54.4

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

14. Applied rewrites 54.4%

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

15. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2024351 
(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)))))))