Midpoint on a great circle

Percentage Accurate: 98.6% → 98.6%

Time: 9.7s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ t_2 := t\_0 \cdot \cos \phi_2\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot t\_3}\\ t_5 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \mathbf{if}\;t\_4 \leq -5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{elif}\;t\_4 \leq -0.02:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(t\_3, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2, \lambda_1\right), \lambda_2, 1\right)}\\ \mathbf{elif}\;t\_4 \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_5 \cdot t\_0}{1 + t\_5 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (cos phi2) t_0))
        (t_2 (* t_0 (cos phi2)))
        (t_3 (cos (- lambda1 lambda2)))
        (t_4 (+ lambda1 (atan2 t_1 (+ (cos phi1) (* (cos phi2) t_3)))))
        (t_5 (fma (* phi2 phi2) -0.5 1.0)))
   (if (<= t_4 -5.0)
     (+ lambda1 (atan2 t_1 (fma (cos lambda1) (cos phi2) (cos phi1))))
     (if (<= t_4 -0.02)
       (atan2 t_2 (fma t_3 (cos phi2) (cos phi1)))
       (if (<= t_4 5e-7)
         (+
          lambda1
          (atan2
           t_1
           (+
            (cos phi1)
            (* (cos phi2) (fma (fma -0.5 lambda2 lambda1) lambda2 1.0)))))
         (if (<= t_4 5.0)
           (atan2 t_2 (fma (cos lambda2) (cos phi2) (cos phi1)))
           (+
            lambda1
            (atan2
             (* t_5 t_0)
             (+
              1.0
              (* t_5 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0)))))))))))))

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 t_2 = t_0 * cos(phi2);
	double t_3 = cos((lambda1 - lambda2));
	double t_4 = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * t_3)));
	double t_5 = fma((phi2 * phi2), -0.5, 1.0);
	double tmp;
	if (t_4 <= -5.0) {
		tmp = lambda1 + atan2(t_1, fma(cos(lambda1), cos(phi2), cos(phi1)));
	} else if (t_4 <= -0.02) {
		tmp = atan2(t_2, fma(t_3, cos(phi2), cos(phi1)));
	} else if (t_4 <= 5e-7) {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0))));
	} else if (t_4 <= 5.0) {
		tmp = atan2(t_2, fma(cos(lambda2), cos(phi2), cos(phi1)));
	} else {
		tmp = lambda1 + atan2((t_5 * t_0), (1.0 + (t_5 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	t_2 = Float64(t_0 * cos(phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	t_4 = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * t_3))))
	t_5 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	tmp = 0.0
	if (t_4 <= -5.0)
		tmp = Float64(lambda1 + atan(t_1, fma(cos(lambda1), cos(phi2), cos(phi1))));
	elseif (t_4 <= -0.02)
		tmp = atan(t_2, fma(t_3, cos(phi2), cos(phi1)));
	elseif (t_4 <= 5e-7)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0)))));
	elseif (t_4 <= 5.0)
		tmp = atan(t_2, fma(cos(lambda2), cos(phi2), cos(phi1)));
	else
		tmp = Float64(lambda1 + atan(Float64(t_5 * t_0), Float64(1.0 + Float64(t_5 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$4, -5.0], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.02], N[ArcTan[t$95$2 / N[(t$95$3 * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e-7], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(-0.5 * lambda2 + lambda1), $MachinePrecision] * lambda2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5.0], N[ArcTan[t$95$2 / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$5 * t$95$0), $MachinePrecision] / N[(1.0 + N[(t$95$5 * N[Sin[N[((-N[(lambda1 - lambda2), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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)))))) < -5

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 78.6

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

   4. Applied rewrites 78.6%

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

   if -5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -0.0200000000000000004

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-cos.f64 42.0

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

   4. Applied rewrites 42.0%

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

   if -0.0200000000000000004 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 4.99999999999999977e-7

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if 4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 5

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-cos.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 42.0

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

   7. Applied rewrites 42.0%

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

   if 5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lower-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift--.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in phi1 around 0

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

       1. sin-+PI/2-rev 61.4

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 61.4

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

   12. Applied rewrites 61.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1} + \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 10^{-25}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          lambda1
          (atan2
           (* (cos phi2) (sin (- lambda2)))
           (+ (cos phi1) (* (cos phi2) (cos (- lambda2))))))))
   (if (<= lambda2 -2e-7)
     t_0
     (if (<= lambda2 1e-25)
       (+
        lambda1
        (atan2
         (* (cos phi2) (sin (- lambda1 lambda2)))
         (fma (cos lambda1) (cos phi2) (cos phi1))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + (cos(phi2) * cos(-lambda2))));
	double tmp;
	if (lambda2 <= -2e-7) {
		tmp = t_0;
	} else if (lambda2 <= 1e-25) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(cos(lambda1), cos(phi2), cos(phi1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(-lambda2))))))
	tmp = 0.0
	if (lambda2 <= -2e-7)
		tmp = t_0;
	elseif (lambda2 <= 1e-25)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(lambda1), cos(phi2), cos(phi1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2e-7], t$95$0, If[LessEqual[lambda2, 1e-25], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.9999999999999999e-7 or 1.00000000000000004e-25 < lambda2

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.9999999999999999e-7 < lambda2 < 1.00000000000000004e-25

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 78.6

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

   4. Applied rewrites 78.6%

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

Alternative 4: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t\_2\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos \phi_1 + \cos \phi_2 \cdot t\_1}\\ t_5 := t\_2 \cdot \cos \phi_2\\ \mathbf{if}\;t\_4 \leq -5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(t\_1, \cos \phi_2, 1\right)}\\ \mathbf{elif}\;t\_4 \leq -0.02:\\ \;\;\;\;\tan^{-1}_* \frac{t\_5}{\mathsf{fma}\left(t\_1, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2, \lambda_1\right), \lambda_2, 1\right)}\\ \mathbf{elif}\;t\_4 \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_5}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot t\_2}{1 + t\_0 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (sin (- lambda1 lambda2)))
        (t_3 (* (cos phi2) t_2))
        (t_4 (+ lambda1 (atan2 t_3 (+ (cos phi1) (* (cos phi2) t_1)))))
        (t_5 (* t_2 (cos phi2))))
   (if (<= t_4 -5.0)
     (+ lambda1 (atan2 t_3 (fma t_1 (cos phi2) 1.0)))
     (if (<= t_4 -0.02)
       (atan2 t_5 (fma t_1 (cos phi2) (cos phi1)))
       (if (<= t_4 5e-7)
         (+
          lambda1
          (atan2
           t_3
           (+
            (cos phi1)
            (* (cos phi2) (fma (fma -0.5 lambda2 lambda1) lambda2 1.0)))))
         (if (<= t_4 5.0)
           (atan2 t_5 (fma (cos lambda2) (cos phi2) (cos phi1)))
           (+
            lambda1
            (atan2
             (* t_0 t_2)
             (+
              1.0
              (* t_0 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0)))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin((lambda1 - lambda2));
	double t_3 = cos(phi2) * t_2;
	double t_4 = lambda1 + atan2(t_3, (cos(phi1) + (cos(phi2) * t_1)));
	double t_5 = t_2 * cos(phi2);
	double tmp;
	if (t_4 <= -5.0) {
		tmp = lambda1 + atan2(t_3, fma(t_1, cos(phi2), 1.0));
	} else if (t_4 <= -0.02) {
		tmp = atan2(t_5, fma(t_1, cos(phi2), cos(phi1)));
	} else if (t_4 <= 5e-7) {
		tmp = lambda1 + atan2(t_3, (cos(phi1) + (cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0))));
	} else if (t_4 <= 5.0) {
		tmp = atan2(t_5, fma(cos(lambda2), cos(phi2), cos(phi1)));
	} else {
		tmp = lambda1 + atan2((t_0 * t_2), (1.0 + (t_0 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = sin(Float64(lambda1 - lambda2))
	t_3 = Float64(cos(phi2) * t_2)
	t_4 = Float64(lambda1 + atan(t_3, Float64(cos(phi1) + Float64(cos(phi2) * t_1))))
	t_5 = Float64(t_2 * cos(phi2))
	tmp = 0.0
	if (t_4 <= -5.0)
		tmp = Float64(lambda1 + atan(t_3, fma(t_1, cos(phi2), 1.0)));
	elseif (t_4 <= -0.02)
		tmp = atan(t_5, fma(t_1, cos(phi2), cos(phi1)));
	elseif (t_4 <= 5e-7)
		tmp = Float64(lambda1 + atan(t_3, Float64(cos(phi1) + Float64(cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0)))));
	elseif (t_4 <= 5.0)
		tmp = atan(t_5, fma(cos(lambda2), cos(phi2), cos(phi1)));
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_2), Float64(1.0 + Float64(t_0 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5.0], N[(lambda1 + N[ArcTan[t$95$3 / N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.02], N[ArcTan[t$95$5 / N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e-7], N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(-0.5 * lambda2 + lambda1), $MachinePrecision] * lambda2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5.0], N[ArcTan[t$95$5 / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$2), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[Sin[N[((-N[(lambda1 - lambda2), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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)))))) < -5

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if -5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -0.0200000000000000004

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-cos.f64 42.0

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

   4. Applied rewrites 42.0%

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

   if -0.0200000000000000004 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 4.99999999999999977e-7

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if 4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 5

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-cos.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 42.0

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

   7. Applied rewrites 42.0%

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

   if 5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lower-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift--.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in phi1 around 0

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

       1. sin-+PI/2-rev 61.4

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 61.4

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

   12. Applied rewrites 61.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1} + \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right) \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t\_2\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos \phi_1 + \cos \phi_2 \cdot t\_1}\\ t_5 := \tan^{-1}_* \frac{t\_2 \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{if}\;t\_4 \leq -5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(t\_1, \cos \phi_2, 1\right)}\\ \mathbf{elif}\;t\_4 \leq -0.02:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2, \lambda_1\right), \lambda_2, 1\right)}\\ \mathbf{elif}\;t\_4 \leq 5:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot t\_2}{1 + t\_0 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (sin (- lambda1 lambda2)))
        (t_3 (* (cos phi2) t_2))
        (t_4 (+ lambda1 (atan2 t_3 (+ (cos phi1) (* (cos phi2) t_1)))))
        (t_5
         (atan2 (* t_2 (cos phi2)) (fma (cos lambda2) (cos phi2) (cos phi1)))))
   (if (<= t_4 -5.0)
     (+ lambda1 (atan2 t_3 (fma t_1 (cos phi2) 1.0)))
     (if (<= t_4 -0.02)
       t_5
       (if (<= t_4 5e-7)
         (+
          lambda1
          (atan2
           t_3
           (+
            (cos phi1)
            (* (cos phi2) (fma (fma -0.5 lambda2 lambda1) lambda2 1.0)))))
         (if (<= t_4 5.0)
           t_5
           (+
            lambda1
            (atan2
             (* t_0 t_2)
             (+
              1.0
              (* t_0 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0)))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin((lambda1 - lambda2));
	double t_3 = cos(phi2) * t_2;
	double t_4 = lambda1 + atan2(t_3, (cos(phi1) + (cos(phi2) * t_1)));
	double t_5 = atan2((t_2 * cos(phi2)), fma(cos(lambda2), cos(phi2), cos(phi1)));
	double tmp;
	if (t_4 <= -5.0) {
		tmp = lambda1 + atan2(t_3, fma(t_1, cos(phi2), 1.0));
	} else if (t_4 <= -0.02) {
		tmp = t_5;
	} else if (t_4 <= 5e-7) {
		tmp = lambda1 + atan2(t_3, (cos(phi1) + (cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0))));
	} else if (t_4 <= 5.0) {
		tmp = t_5;
	} else {
		tmp = lambda1 + atan2((t_0 * t_2), (1.0 + (t_0 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = sin(Float64(lambda1 - lambda2))
	t_3 = Float64(cos(phi2) * t_2)
	t_4 = Float64(lambda1 + atan(t_3, Float64(cos(phi1) + Float64(cos(phi2) * t_1))))
	t_5 = atan(Float64(t_2 * cos(phi2)), fma(cos(lambda2), cos(phi2), cos(phi1)))
	tmp = 0.0
	if (t_4 <= -5.0)
		tmp = Float64(lambda1 + atan(t_3, fma(t_1, cos(phi2), 1.0)));
	elseif (t_4 <= -0.02)
		tmp = t_5;
	elseif (t_4 <= 5e-7)
		tmp = Float64(lambda1 + atan(t_3, Float64(cos(phi1) + Float64(cos(phi2) * fma(fma(-0.5, lambda2, lambda1), lambda2, 1.0)))));
	elseif (t_4 <= 5.0)
		tmp = t_5;
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_2), Float64(1.0 + Float64(t_0 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(t$95$2 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -5.0], N[(lambda1 + N[ArcTan[t$95$3 / N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.02], t$95$5, If[LessEqual[t$95$4, 5e-7], N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(-0.5 * lambda2 + lambda1), $MachinePrecision] * lambda2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5.0], t$95$5, N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$2), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[Sin[N[((-N[(lambda1 - lambda2), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 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)))))) < -5

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if -5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -0.0200000000000000004 or 4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 5

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-cos.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 42.0

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

   7. Applied rewrites 42.0%

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

   if -0.0200000000000000004 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 4.99999999999999977e-7

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if 5 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lower-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift--.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in phi1 around 0

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

       1. sin-+PI/2-rev 61.4

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 61.4

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

   12. Applied rewrites 61.4%

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

Alternative 6: 87.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(\phi_2 \cdot \phi_2\right) - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ \mathbf{if}\;\phi_2 \leq 0.45:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot t\_0}{\cos \phi_1 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\lambda_2, \lambda_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1
         (fma
          (-
           (*
            (fma (* phi2 phi2) -0.001388888888888889 0.041666666666666664)
            (* phi2 phi2))
           0.5)
          (* phi2 phi2)
          1.0)))
   (if (<= phi2 0.45)
     (+
      lambda1
      (atan2 (* t_1 t_0) (+ (cos phi1) (* t_1 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_0)
       (+ (cos phi1) (* (cos phi2) (fma lambda2 lambda1 1.0))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = fma(((fma((phi2 * phi2), -0.001388888888888889, 0.041666666666666664) * (phi2 * phi2)) - 0.5), (phi2 * phi2), 1.0);
	double tmp;
	if (phi2 <= 0.45) {
		tmp = lambda1 + atan2((t_1 * t_0), (cos(phi1) + (t_1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi1) + (cos(phi2) * fma(lambda2, lambda1, 1.0))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = fma(Float64(Float64(fma(Float64(phi2 * phi2), -0.001388888888888889, 0.041666666666666664) * Float64(phi2 * phi2)) - 0.5), Float64(phi2 * phi2), 1.0)
	tmp = 0.0
	if (phi2 <= 0.45)
		tmp = Float64(lambda1 + atan(Float64(t_1 * t_0), Float64(cos(phi1) + Float64(t_1 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi1) + Float64(cos(phi2) * fma(lambda2, lambda1, 1.0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 0.45], N[(lambda1 + N[ArcTan[N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(lambda2 * lambda1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.450000000000000011

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if 0.450000000000000011 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.8

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

   7. Applied rewrites 76.8%

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

Alternative 7: 86.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0))
        (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.9995)
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (+ (cos phi1) (* (cos phi2) (fma lambda2 lambda1 1.0)))))
     (+
      lambda1
      (atan2 (* t_0 t_1) (+ (cos phi1) (* t_0 (cos (- lambda1 lambda2)))))))))

C

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.9995)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(cos(phi1) + Float64(cos(phi2) * fma(lambda2, lambda1, 1.0)))));
	else
		tmp = Float64(lambda1 + atan(Float64(t_0 * t_1), Float64(cos(phi1) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9995], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(lambda2 * lambda1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(t$95$0 * t$95$1), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 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. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sin-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if 0.99950000000000006 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

Alternative 8: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.041666666666666664 - 0.5, \phi_2 \cdot \phi_2, 1\right)\\ \mathbf{if}\;\phi_2 \leq 0.235:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2 \cdot t\_0}{\cos \phi_1 + t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(t\_1, \cos \phi_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2
         (fma
          (- (* (* phi2 phi2) 0.041666666666666664) 0.5)
          (* phi2 phi2)
          1.0)))
   (if (<= phi2 0.235)
     (+ lambda1 (atan2 (* t_2 t_0) (+ (cos phi1) (* t_2 t_1))))
     (+ lambda1 (atan2 (* (cos phi2) t_0) (fma t_1 (cos phi2) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = fma((((phi2 * phi2) * 0.041666666666666664) - 0.5), (phi2 * phi2), 1.0);
	double tmp;
	if (phi2 <= 0.235) {
		tmp = lambda1 + atan2((t_2 * t_0), (cos(phi1) + (t_2 * t_1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), fma(t_1, cos(phi2), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = fma(Float64(Float64(Float64(phi2 * phi2) * 0.041666666666666664) - 0.5), Float64(phi2 * phi2), 1.0)
	tmp = 0.0
	if (phi2 <= 0.235)
		tmp = Float64(lambda1 + atan(Float64(t_2 * t_0), Float64(cos(phi1) + Float64(t_2 * t_1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), fma(t_1, cos(phi2), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[phi2, 0.235], N[(lambda1 + N[ArcTan[N[(t$95$2 * t$95$0), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.23499999999999999

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if 0.23499999999999999 < phi2

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 9: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.87:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.87)
     (+ 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 (cos(phi2) <= 0.87) {
		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 (cos(phi2) <= 0.87)
		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[N[Cos[phi2], $MachinePrecision], 0.87], 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 (cos.f64 phi2) < 0.869999999999999996

   1. Initial program 98.6%

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

   2. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if 0.869999999999999996 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 78.1

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

   4. Applied rewrites 78.1%

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

Alternative 10: 78.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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((lambda1 - lambda2)) + cos(phi1)))
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((lambda1 - lambda2)) + Math.cos(phi1)));
}

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) + 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[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-cos.f64 78.1

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

4. Applied rewrites 78.1%

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

Alternative 11: 76.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 75.7

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

4. Applied rewrites 75.7%

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

5. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.3

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

7. Applied rewrites 76.3%

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

Alternative 12: 75.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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(cos(Float64(lambda1 - lambda2)) + cos(phi1))))
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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 75.7

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

4. Applied rewrites 75.7%

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

5. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.3

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

7. Applied rewrites 76.3%

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

8. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-cos.f64 75.2

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

10. Applied rewrites 75.2%

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

Alternative 13: 66.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := t\_0 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.99:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 + t\_0 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\left(\phi_1 \cdot \phi_1\right) \cdot 0.041666666666666664 - 0.5, \phi_1 \cdot \phi_1, 1\right) + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (* t_0 (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.99)
     (+
      lambda1
      (atan2 t_1 (+ 1.0 (* t_0 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0)))))))
     (+
      lambda1
      (atan2
       t_1
       (+
        (fma (- (* (* phi1 phi1) 0.041666666666666664) 0.5) (* phi1 phi1) 1.0)
        (* t_0 (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = t_0 * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.99) {
		tmp = lambda1 + atan2(t_1, (1.0 + (t_0 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
	} else {
		tmp = lambda1 + atan2(t_1, (fma((((phi1 * phi1) * 0.041666666666666664) - 0.5), (phi1 * phi1), 1.0) + (t_0 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = Float64(t_0 * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.99)
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 + Float64(t_0 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(fma(Float64(Float64(Float64(phi1 * phi1) * 0.041666666666666664) - 0.5), Float64(phi1 * phi1), 1.0) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.99], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 + N[(t$95$0 * N[Sin[N[((-N[(lambda1 - lambda2), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(N[(N[(phi1 * phi1), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.98999999999999999

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lower-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift--.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in phi1 around 0

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

       1. sin-+PI/2-rev 61.4

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 61.4

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

   12. Applied rewrites 61.4%

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

   if 0.98999999999999999 < (cos.f64 phi1)

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in phi1 around 0

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

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

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. sin-sum-rev N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   10. Applied rewrites 61.2%

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

Alternative 14: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ t_1 := t\_0 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 9 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 + t\_0 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0))
        (t_1 (* t_0 (sin (- lambda1 lambda2)))))
   (if (<= phi1 9e-22)
     (+
      lambda1
      (atan2
       t_1
       (+ (fma -0.5 (* phi1 phi1) 1.0) (* t_0 (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2
       t_1
       (+ 1.0 (* t_0 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	double t_1 = t_0 * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 9e-22) {
		tmp = lambda1 + atan2(t_1, (fma(-0.5, (phi1 * phi1), 1.0) + (t_0 * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2(t_1, (1.0 + (t_0 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	t_1 = Float64(t_0 * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 9e-22)
		tmp = Float64(lambda1 + atan(t_1, Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 + Float64(t_0 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 9e-22], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 + N[(t$95$0 * N[Sin[N[((-N[(lambda1 - lambda2), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 8.99999999999999973e-22

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in phi1 around 0

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

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

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. sin-sum-rev N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   10. Applied rewrites 61.2%

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

   11. Taylor expanded in phi1 around 0

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

   13. Applied rewrites 63.8%

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

   if 8.99999999999999973e-22 < phi1

   1. Initial program 98.6%

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

   2. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lower-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift--.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in phi1 around 0

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

       1. sin-+PI/2-rev 61.4

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

       2. lift-/.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. sin-sum-rev N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-/.f64 61.4

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

   12. Applied rewrites 61.4%

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

Alternative 15: 61.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)\\ \lambda_1 + \tan^{-1}_* \frac{t\_0 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + t\_0 \cdot \sin \left(\left(-\left(\lambda_1 - \lambda_2\right)\right) + \frac{\pi}{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* phi2 phi2) -0.5 1.0)))
   (+
    lambda1
    (atan2
     (* t_0 (sin (- lambda1 lambda2)))
     (+ 1.0 (* t_0 (sin (+ (- (- lambda1 lambda2)) (/ PI 2.0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi2 * phi2), -0.5, 1.0);
	return lambda1 + atan2((t_0 * sin((lambda1 - lambda2))), (1.0 + (t_0 * sin((-(lambda1 - lambda2) + (((double) M_PI) / 2.0))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi2 * phi2), -0.5, 1.0)
	return Float64(lambda1 + atan(Float64(t_0 * sin(Float64(lambda1 - lambda2))), Float64(1.0 + Float64(t_0 * sin(Float64(Float64(-Float64(lambda1 - lambda2)) + Float64(pi / 2.0)))))))
end

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 75.7

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

4. Applied rewrites 75.7%

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

5. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.3

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

7. Applied rewrites 76.3%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. lower-sin.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift--.f64 66.4

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

9. Applied rewrites 66.4%

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

10. Taylor expanded in phi1 around 0

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

    1. sin-+PI/2-rev 61.4

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

    2. lift-/.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. sin-sum-rev N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-/.f64 61.4

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

12. Applied rewrites 61.4%

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

Alternative 16: 52.1% accurate, 48.6× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2) :precision binary64 (* 1.0 lambda1))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return 1.0 * lambda1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return 1.0 * lambda1;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return 1.0 * lambda1

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(1.0 * lambda1)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = 1.0 * lambda1;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(1.0 * lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in lambda1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in lambda1 around inf

   \[\leadsto 1 \cdot \lambda_1 \]
6. Step-by-step derivation

7. Applied rewrites 52.1%

   \[\leadsto 1 \cdot \lambda_1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025141 
(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)))))))