Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.7% → 96.2%

Time: 17.2s

Alternatives: 15

Speedup: 3.0×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5 \cdot 10^{-196}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 7 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5e-196)
   (* R (hypot (* lambda1 (cos (* phi1 0.5))) (- phi1 phi2)))
   (if (<= lambda2 7e+156)
     (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))
     (* (* R lambda2) (sqrt (+ 0.5 (* 0.5 (cos (+ phi1 phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5e-196) {
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	} else if (lambda2 <= 7e+156) {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos((phi1 + phi2)))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5e-196) {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else if (lambda2 <= 7e+156) {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = (R * lambda2) * Math.sqrt((0.5 + (0.5 * Math.cos((phi1 + phi2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5e-196:
		tmp = R * math.hypot((lambda1 * math.cos((phi1 * 0.5))), (phi1 - phi2))
	elif lambda2 <= 7e+156:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	else:
		tmp = (R * lambda2) * math.sqrt((0.5 + (0.5 * math.cos((phi1 + phi2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5e-196)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	elseif (lambda2 <= 7e+156)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	else
		tmp = Float64(Float64(R * lambda2) * sqrt(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5e-196)
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	elseif (lambda2 <= 7e+156)
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	else
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos((phi1 + phi2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5e-196], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 7e+156], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 5.0000000000000005e-196

   1. Initial program 56.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda1 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right), \lambda_1\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right), \lambda_1\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-lowering-*.f64 75.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \lambda_1\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   10. Simplified 75.2%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1}, \phi_1 - \phi_2\right) \]

   if 5.0000000000000005e-196 < lambda2 < 7.0000000000000006e156

   1. Initial program 63.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 95.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 95.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 92.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 92.0%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   if 7.0000000000000006e156 < lambda2

   1. Initial program 45.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right), \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\phi_1 - \phi_2\right), \left(\phi_1 - \phi_2\right)\right), \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \left(\phi_1 - \phi_2\right)\right), \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 45.8%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

   5. Taylor expanded in lambda2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\left(\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\phi_1 + \phi_2\right)}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \left(\sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\phi_1 + \phi_2\right)}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(\phi_1 + \phi_2\right)\right)\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\phi_1 + \phi_2\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 81.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\phi_2, \phi_1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 81.8%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5 \cdot 10^{-196}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 7 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e-29)
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-29) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-29) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-29:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-29)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e-29)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e-29], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.99999999999999986e-29

   1. Initial program 58.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 93.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   if 4.99999999999999986e-29 < phi2

   1. Initial program 55.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 94.4%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-199}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1e-199)
   (* R (hypot (* lambda1 (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-199) {
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-199) {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1e-199:
		tmp = R * math.hypot((lambda1 * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1e-199)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1e-199)
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1e-199], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.99999999999999982e-200

   1. Initial program 57.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 97.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda1 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right), \lambda_1\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right), \lambda_1\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \lambda_1\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   10. Simplified 75.0%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1}, \phi_1 - \phi_2\right) \]

   if 9.99999999999999982e-200 < lambda2

   1. Initial program 57.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 97.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 90.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 85.7%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-199}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. *-lowering-*.f64 90.5%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

7. Simplified 90.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

8. Final simplification 90.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right) \]
9. Add Preprocessing

Alternative 6: 71.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{+40}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.5e+40)
   (* R (hypot (- lambda1 lambda2) phi1))
   (* R (hypot (- lambda1 lambda2) phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.5e+40) {
		tmp = R * hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * hypot((lambda1 - lambda2), phi2);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.5e+40) {
		tmp = R * Math.hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), phi2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.5e+40:
		tmp = R * math.hypot((lambda1 - lambda2), phi1)
	else:
		tmp = R * math.hypot((lambda1 - lambda2), phi2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.5e+40)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), phi1));
	else
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.5e+40)
		tmp = R * hypot((lambda1 - lambda2), phi1);
	else
		tmp = R * hypot((lambda1 - lambda2), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.5e+40], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.50000000000000032e40

   1. Initial program 45.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 84.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 84.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + {\phi_1}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_1}\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\lambda_1 - \lambda_2, \color{blue}{\phi_1}\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\lambda_1 - \lambda_2\right), \color{blue}{\phi_1}\right)\right) \]

      7. --lowering--.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \phi_1\right)\right) \]

   10. Simplified 77.4%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)} \]

   if -4.50000000000000032e40 < phi1

   1. Initial program 60.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 89.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 89.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + {\phi_2}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_2}\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\lambda_1 - \lambda_2, \color{blue}{\phi_2}\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\lambda_1 - \lambda_2\right), \color{blue}{\phi_2}\right)\right) \]

      7. --lowering--.f64 72.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \phi_2\right)\right) \]

   10. Simplified 72.9%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{+52}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.4e+52)
   (* R (hypot (- lambda1 lambda2) phi1))
   (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.4e+52) {
		tmp = R * hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.4e+52) {
		tmp = R * Math.hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.4e+52:
		tmp = R * math.hypot((lambda1 - lambda2), phi1)
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.4e+52)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), phi1));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.4e+52)
		tmp = R * hypot((lambda1 - lambda2), phi1);
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.4e+52], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.4e52

   1. Initial program 56.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 91.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + {\phi_1}^{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_1}\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\lambda_1 - \lambda_2, \color{blue}{\phi_1}\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\lambda_1 - \lambda_2\right), \color{blue}{\phi_1}\right)\right) \]

      7. --lowering--.f64 70.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \phi_1\right)\right) \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)} \]

   if 3.4e52 < phi2

   1. Initial program 58.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_1}{\phi_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 - \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_1}{\phi_2}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 80.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   5. Simplified 80.4%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (- lambda1 lambda2) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((lambda1 - lambda2), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((lambda1 - lambda2), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. *-lowering-*.f64 90.5%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

7. Simplified 90.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
9. Step-by-step derivation

   1. --lowering--.f64 85.2%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

10. Simplified 85.2%

    \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]
11. Add Preprocessing

Alternative 9: 29.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7 \cdot 10^{-117}:\\ \;\;\;\;0 - R \cdot \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -7e-117)
   (- 0.0 (* R lambda1))
   (if (<= lambda2 1.9e+130)
     (* phi1 (- (/ (* R phi2) phi1) R))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7e-117) {
		tmp = 0.0 - (R * lambda1);
	} else if (lambda2 <= 1.9e+130) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-7d-117)) then
        tmp = 0.0d0 - (r * lambda1)
    else if (lambda2 <= 1.9d+130) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7e-117) {
		tmp = 0.0 - (R * lambda1);
	} else if (lambda2 <= 1.9e+130) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -7e-117:
		tmp = 0.0 - (R * lambda1)
	elif lambda2 <= 1.9e+130:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -7e-117)
		tmp = Float64(0.0 - Float64(R * lambda1));
	elseif (lambda2 <= 1.9e+130)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -7e-117)
		tmp = 0.0 - (R * lambda1);
	elseif (lambda2 <= 1.9e+130)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -7e-117], N[(0.0 - N[(R * lambda1), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.9e+130], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -6.9999999999999997e-117

   1. Initial program 51.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 89.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 89.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(R \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(R \cdot \lambda_1\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \]

      8. *-lowering-*.f64 11.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right)\right) \]

   10. Simplified 11.4%

       \[\leadsto \color{blue}{0 - \left(R \cdot \lambda_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(R \cdot \lambda_1\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{R \cdot \lambda_1} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \lambda_1\right)}\right) \]

       4. *-lowering-*.f64 9.5%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\lambda_1}\right)\right) \]

   13. Simplified 9.5%

       \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

   if -6.9999999999999997e-117 < lambda2 < 1.9000000000000001e130

   1. Initial program 64.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right) \cdot \phi_1\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\phi_1\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right) \cdot \left(-1 \cdot \color{blue}{\phi_1}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right), \color{blue}{\left(-1 \cdot \phi_1\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)\right), \left(-1 \cdot \phi_1\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R - \frac{R \cdot \phi_2}{\phi_1}\right), \left(\color{blue}{-1} \cdot \phi_1\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \left(\frac{R \cdot \phi_2}{\phi_1}\right)\right), \left(\color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_2\right), \phi_1\right)\right), \left(-1 \cdot \phi_1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right), \left(-1 \cdot \phi_1\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right), \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right), \left(0 - \color{blue}{\phi_1}\right)\right) \]

      13. --lowering--.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\phi_1}\right)\right) \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\left(R - \frac{R \cdot \phi_2}{\phi_1}\right) \cdot \left(0 - \phi_1\right)} \]

   if 1.9000000000000001e130 < lambda2

   1. Initial program 48.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 93.8%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 85.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \color{blue}{\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}}{\lambda_2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \color{blue}{\lambda_2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      11. *-lowering-*.f64 51.7%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

   10. Simplified 51.7%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) - \frac{\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1}{\lambda_2}\right)\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 - \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       4. /-lowering-/.f64 60.8%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 60.8%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7 \cdot 10^{-117}:\\ \;\;\;\;0 - R \cdot \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7 \cdot 10^{-117}:\\ \;\;\;\;0 - R \cdot \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 3.9 \cdot 10^{+128}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -7e-117)
   (- 0.0 (* R lambda1))
   (if (<= lambda2 3.9e+128)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7e-117) {
		tmp = 0.0 - (R * lambda1);
	} else if (lambda2 <= 3.9e+128) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-7d-117)) then
        tmp = 0.0d0 - (r * lambda1)
    else if (lambda2 <= 3.9d+128) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7e-117) {
		tmp = 0.0 - (R * lambda1);
	} else if (lambda2 <= 3.9e+128) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -7e-117:
		tmp = 0.0 - (R * lambda1)
	elif lambda2 <= 3.9e+128:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -7e-117)
		tmp = Float64(0.0 - Float64(R * lambda1));
	elseif (lambda2 <= 3.9e+128)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -7e-117)
		tmp = 0.0 - (R * lambda1);
	elseif (lambda2 <= 3.9e+128)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -7e-117], N[(0.0 - N[(R * lambda1), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 3.9e+128], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -6.9999999999999997e-117

   1. Initial program 51.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 89.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 89.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(R \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(R \cdot \lambda_1\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \]

      8. *-lowering-*.f64 11.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_1\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right)\right) \]

   10. Simplified 11.4%

       \[\leadsto \color{blue}{0 - \left(R \cdot \lambda_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(R \cdot \lambda_1\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{R \cdot \lambda_1} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \lambda_1\right)}\right) \]

       4. *-lowering-*.f64 9.5%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\lambda_1}\right)\right) \]

   13. Simplified 9.5%

       \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

   if -6.9999999999999997e-117 < lambda2 < 3.8999999999999997e128

   1. Initial program 64.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-lowering-*.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_1\right), \phi_2\right)\right)\right) \]

   5. Simplified 41.6%

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)} \]

   if 3.8999999999999997e128 < lambda2

   1. Initial program 48.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 93.8%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 85.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \color{blue}{\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}}{\lambda_2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \color{blue}{\lambda_2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      11. *-lowering-*.f64 51.7%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

   10. Simplified 51.7%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) - \frac{\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1}{\lambda_2}\right)\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 - \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       4. /-lowering-/.f64 60.8%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 60.8%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 32.1% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.3 \cdot 10^{-279}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.3e-279)
   (* R (- 0.0 phi1))
   (if (<= phi2 2.05e-7)
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.3e-279) {
		tmp = R * (0.0 - phi1);
	} else if (phi2 <= 2.05e-7) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.3d-279) then
        tmp = r * (0.0d0 - phi1)
    else if (phi2 <= 2.05d-7) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.3e-279) {
		tmp = R * (0.0 - phi1);
	} else if (phi2 <= 2.05e-7) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.3e-279:
		tmp = R * (0.0 - phi1)
	elif phi2 <= 2.05e-7:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.3e-279)
		tmp = Float64(R * Float64(0.0 - phi1));
	elseif (phi2 <= 2.05e-7)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.3e-279)
		tmp = R * (0.0 - phi1);
	elseif (phi2 <= 2.05e-7)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.3e-279], N[(R * N[(0.0 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.05e-7], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.29999999999999995e-279

   1. Initial program 56.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 21.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   5. Simplified 21.1%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot R\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \color{blue}{R} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\phi_1\right)\right), \color{blue}{R}\right) \]

      5. neg-lowering-neg.f64 21.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\phi_1\right), R\right) \]

   7. Applied egg-rr 21.1%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if 2.29999999999999995e-279 < phi2 < 2.05e-7

   1. Initial program 61.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 88.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \color{blue}{\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}}{\lambda_2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \color{blue}{\lambda_2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      11. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

   10. Simplified 28.5%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) - \frac{\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1}{\lambda_2}\right)\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 - \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       4. /-lowering-/.f64 28.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 28.5%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]

   if 2.05e-7 < phi2

   1. Initial program 54.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_1}{\phi_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 - \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_1}{\phi_2}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 70.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   5. Simplified 70.0%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.3 \cdot 10^{-279}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.16 \cdot 10^{+49}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.5 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.16e+49)
   (* R (- 0.0 phi1))
   (if (<= phi1 2.5e-246)
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.16e+49) {
		tmp = R * (0.0 - phi1);
	} else if (phi1 <= 2.5e-246) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.16d+49)) then
        tmp = r * (0.0d0 - phi1)
    else if (phi1 <= 2.5d-246) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.16e+49) {
		tmp = R * (0.0 - phi1);
	} else if (phi1 <= 2.5e-246) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.16e+49:
		tmp = R * (0.0 - phi1)
	elif phi1 <= 2.5e-246:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.16e+49)
		tmp = Float64(R * Float64(0.0 - phi1));
	elseif (phi1 <= 2.5e-246)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.16e+49)
		tmp = R * (0.0 - phi1);
	elseif (phi1 <= 2.5e-246)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.16e+49], N[(R * N[(0.0 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.5e-246], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.16000000000000003e49

   1. Initial program 45.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   5. Simplified 70.9%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot R\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \color{blue}{R} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\phi_1\right)\right), \color{blue}{R}\right) \]

      5. neg-lowering-neg.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\phi_1\right), R\right) \]

   7. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -2.16000000000000003e49 < phi1 < 2.4999999999999998e-246

   1. Initial program 68.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 98.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \color{blue}{\left(\frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}}{\lambda_2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{\color{blue}{\lambda_1} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}{\lambda_2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \color{blue}{\lambda_2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

      11. *-lowering-*.f64 29.0%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \lambda_1\right), \lambda_2\right)\right)\right)\right) \]

   10. Simplified 29.0%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) - \frac{\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1}{\lambda_2}\right)\right)} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 - \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       4. /-lowering-/.f64 29.1%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 29.1%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]

   if 2.4999999999999998e-246 < phi1

   1. Initial program 53.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 22.7%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   5. Simplified 22.7%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.16 \cdot 10^{+49}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.5 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.1% accurate, 32.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2e+20) (* R (- 0.0 phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2e+20) {
		tmp = R * (0.0 - phi1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2d+20)) then
        tmp = r * (0.0d0 - phi1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2e+20) {
		tmp = R * (0.0 - phi1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2e+20:
		tmp = R * (0.0 - phi1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2e+20)
		tmp = Float64(R * Float64(0.0 - phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2e+20)
		tmp = R * (0.0 - phi1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2e+20], N[(R * N[(0.0 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2e20

   1. Initial program 44.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 65.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot R\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \color{blue}{R} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\phi_1\right)\right), \color{blue}{R}\right) \]

      5. neg-lowering-neg.f64 65.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\phi_1\right), R\right) \]

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -2e20 < phi1

   1. Initial program 61.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 25.2%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   5. Simplified 25.2%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \left(0 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.2 \cdot 10^{+52}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.2e+52) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.2e+52) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 6.2d+52) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.2e+52) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.2e+52:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.2e+52)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.2e+52)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.2e+52], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.2e52

   1. Initial program 56.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 91.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 17.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 17.7%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 18.2%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 18.2%

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]

   if 6.2e52 < phi2

   1. Initial program 58.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 14.2% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda2), $MachinePrecision]

Derivation

1. Initial program 57.1%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. *-lowering-*.f64 90.5%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

7. Simplified 90.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

8. Taylor expanded in lambda2 around inf

   \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

   5. *-lowering-*.f64 15.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

10. Simplified 15.5%

    \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

11. Taylor expanded in phi1 around 0

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
12. Step-by-step derivation

    1. *-lowering-*.f64 15.1%

       \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

13. Simplified 15.1%

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))