Midpoint on a great circle

Percentage Accurate: 98.5% → 98.5%

Time: 20.4s

Alternatives: 18

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

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

real(8) function code(lambda1, lambda2, phi1, phi2)
    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: 98.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

Alternative 2: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.96999999999999997

   1. Initial program 98.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if 0.96999999999999997 < (cos.f64 phi1)

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 3: 87.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.96999999999999997

   1. Initial program 98.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 79.8

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

   8. Applied rewrites 79.8%

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

   if 0.96999999999999997 < (cos.f64 phi1)

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.94:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.94)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (fma (cos lambda2) (cos phi2) 1.0)))
     (+
      lambda1
      (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.94) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), fma(cos(lambda2), cos(phi2), 1.0));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.94)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), fma(cos(lambda2), cos(phi2), 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.94], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.93999999999999995

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 78.1

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in lambda1 around 0

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

   10. Applied rewrites 77.5%

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

   if 0.93999999999999995 < (cos.f64 phi2)

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 5: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.896:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.896)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (fma (cos lambda1) (cos phi2) 1.0)))
     (+
      lambda1
      (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.896) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), fma(cos(lambda1), cos(phi2), 1.0));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.896)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), fma(cos(lambda1), cos(phi2), 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.896], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.896000000000000019

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in lambda2 around 0

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

   10. Applied rewrites 69.8%

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

   if 0.896000000000000019 < (cos.f64 phi2)

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.04:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 + \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (cos phi2) -0.04)
     (+
      lambda1
      (atan2
       (* (cos phi2) (sin (- lambda2)))
       (+ t_0 (fma phi1 (* phi1 -0.5) 1.0))))
     (+
      lambda1
      (atan2 (sin (- lambda1 lambda2)) (+ (cos phi1) (* (cos phi2) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= -0.04) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (t_0 + fma(phi1, (phi1 * -0.5), 1.0)));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + (cos(phi2) * t_0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= -0.04)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 + fma(phi1, Float64(phi1 * -0.5), 1.0))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.04], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.0400000000000000008

   1. Initial program 99.1%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 49.4

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-neg.f64 55.9

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

   11. Applied rewrites 55.9%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 60.7%

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

   if -0.0400000000000000008 < (cos.f64 phi2)

   1. Initial program 99.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 88.8

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

   5. Applied rewrites 88.8%

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

Alternative 7: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 14500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 14500000.0)
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (fma (cos (- lambda1 lambda2)) (fma -0.5 (* phi2 phi2) 1.0) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda2)))
     (fma (cos phi2) (cos lambda2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 14500000.0) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(cos((lambda1 - lambda2)), fma(-0.5, (phi2 * phi2), 1.0), cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), fma(cos(phi2), cos(lambda2), cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 14500000.0)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(Float64(lambda1 - lambda2)), fma(-0.5, Float64(phi2 * phi2), 1.0), cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), fma(cos(phi2), cos(lambda2), cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 14500000.0], 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[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.45e7

   1. Initial program 99.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if 1.45e7 < phi2

   1. Initial program 99.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 55.1

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-neg.f64 58.5

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

   11. Applied rewrites 58.5%

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

   12. Taylor expanded in lambda1 around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. cos-neg N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-cos.f64 90.3

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

   14. Applied rewrites 90.3%

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

Alternative 8: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.04:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi2) -0.04)
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda2)))
     (+ (cos (- lambda1 lambda2)) (fma phi1 (* phi1 -0.5) 1.0))))
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (fma (cos phi2) (cos lambda2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi2) <= -0.04) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos((lambda1 - lambda2)) + fma(phi1, (phi1 * -0.5), 1.0)));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), fma(cos(phi2), cos(lambda2), cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi2) <= -0.04)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(Float64(lambda1 - lambda2)) + fma(phi1, Float64(phi1 * -0.5), 1.0))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), fma(cos(phi2), cos(lambda2), cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.04], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.0400000000000000008

   1. Initial program 99.1%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 49.4

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-neg.f64 55.9

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

   11. Applied rewrites 55.9%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 60.7%

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

   if -0.0400000000000000008 < (cos.f64 phi2)

   1. Initial program 99.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 88.8

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

   8. Applied rewrites 88.8%

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

Alternative 9: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 10: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.978:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 + \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.978)
     (+
      lambda1
      (atan2
       (* (cos phi2) (sin (- lambda2)))
       (+ t_0 (fma phi1 (* phi1 -0.5) 1.0))))
     (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.978) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (t_0 + fma(phi1, (phi1 * -0.5), 1.0)));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.978)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 + fma(phi1, Float64(phi1 * -0.5), 1.0))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.978], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.97799999999999998

   1. Initial program 98.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-neg.f64 59.8

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

   11. Applied rewrites 59.8%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 63.9%

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

   if 0.97799999999999998 < (cos.f64 phi2)

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 11: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.84:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), t\_0, \cos \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.84)
     (+ lambda1 (atan2 (* (cos phi2) t_1) (+ t_0 1.0)))
     (+
      lambda1
      (atan2 t_1 (fma (fma -0.5 (* phi2 phi2) 1.0) t_0 (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.84) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), (t_0 + 1.0));
	} else {
		tmp = lambda1 + atan2(t_1, fma(fma(-0.5, (phi2 * phi2), 1.0), t_0, cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.84)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), Float64(t_0 + 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(fma(-0.5, Float64(phi2 * phi2), 1.0), t_0, cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.84], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.839999999999999969

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 78.1

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 59.4%

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

   if 0.839999999999999969 < (cos.f64 phi2)

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 95.1

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

   8. Applied rewrites 95.1%

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

Alternative 12: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 42000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) + \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 42000000.0)
   (+ lambda1 (atan2 (sin lambda1) (+ (cos phi1) (cos lambda2))))
   (+
    lambda1
    (atan2
     (sin lambda1)
     (+ (cos (- lambda1 lambda2)) (fma phi1 (* phi1 -0.5) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 42000000.0) {
		tmp = lambda1 + atan2(sin(lambda1), (cos(phi1) + cos(lambda2)));
	} else {
		tmp = lambda1 + atan2(sin(lambda1), (cos((lambda1 - lambda2)) + fma(phi1, (phi1 * -0.5), 1.0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 42000000.0)
		tmp = Float64(lambda1 + atan(sin(lambda1), Float64(cos(phi1) + cos(lambda2))));
	else
		tmp = Float64(lambda1 + atan(sin(lambda1), Float64(cos(Float64(lambda1 - lambda2)) + fma(phi1, Float64(phi1 * -0.5), 1.0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 42000000.0], N[(lambda1 + N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.2e7

   1. Initial program 99.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 85.1

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

   8. Applied rewrites 85.1%

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

   9. Taylor expanded in lambda2 around 0

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

   11. Applied rewrites 62.8%

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

   12. Taylor expanded in lambda1 around 0

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

   14. Applied rewrites 62.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 + \cos \lambda_2} \]

   if 4.2e7 < phi2

   1. Initial program 99.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 55.1

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in lambda2 around 0

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

   11. Applied rewrites 53.9%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 59.5%

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

Alternative 13: 76.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.3

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

8. Applied rewrites 78.3%

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

Alternative 14: 76.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.3

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

8. Applied rewrites 78.3%

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

9. Taylor expanded in lambda1 around 0

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

11. Applied rewrites 78.3%

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

Alternative 15: 67.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.3

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

8. Applied rewrites 78.3%

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

9. Taylor expanded in lambda2 around 0

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

11. Applied rewrites 69.2%

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

Alternative 16: 56.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.3

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

8. Applied rewrites 78.3%

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

9. Taylor expanded in lambda2 around 0

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

11. Applied rewrites 60.8%

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

12. Taylor expanded in phi1 around 0

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

14. Applied rewrites 60.2%

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

Alternative 17: 55.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 78.3

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

8. Applied rewrites 78.3%

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

9. Taylor expanded in lambda2 around 0

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

11. Applied rewrites 60.8%

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

12. Taylor expanded in phi1 around 0

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

14. Applied rewrites 57.7%

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

15. Final simplification 57.7%

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

Alternative 18: 52.7% accurate, 104.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in lambda1 around inf

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in lambda1 around inf

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

12. Applied rewrites 54.4%

    \[\leadsto 1 \cdot \lambda_1 \]

13. Final simplification 54.4%

    \[\leadsto \lambda_1 \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))))))