Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.8% → 95.8%

Time: 16.7s

Alternatives: 13

Speedup: 3.0×

• 30.8% 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: 59.8% 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: 95.8% 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 62.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

Alternative 2: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 950000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 950000.0)
   (* 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 <= 950000.0) {
		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 <= 950000.0) {
		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 <= 950000.0:
		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 <= 950000.0)
		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 <= 950000.0)
		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, 950000.0], 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 < 9.5e5

   1. Initial program 62.4%

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

      1. hypot-define 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in phi2 around 0 90.2%

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

      1. *-commutative 90.2%

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

   7. Simplified 90.2%

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

   if 9.5e5 < phi2

   1. Initial program 62.2%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in phi1 around 0 96.3%

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

      1. *-commutative 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 950000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_1 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1.02 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t\_0, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(t\_0 \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* phi1 0.5))))
   (if (<= lambda1 -1.02e-25)
     (* R (hypot (* lambda1 t_0) (- phi1 phi2)))
     (* R (hypot (* t_0 (- lambda2)) (- phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi1 * 0.5));
	double tmp;
	if (lambda1 <= -1.02e-25) {
		tmp = R * hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R * hypot((t_0 * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((phi1 * 0.5));
	double tmp;
	if (lambda1 <= -1.02e-25) {
		tmp = R * Math.hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((t_0 * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((phi1 * 0.5))
	tmp = 0
	if lambda1 <= -1.02e-25:
		tmp = R * math.hypot((lambda1 * t_0), (phi1 - phi2))
	else:
		tmp = R * math.hypot((t_0 * -lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi1 * 0.5))
	tmp = 0.0
	if (lambda1 <= -1.02e-25)
		tmp = Float64(R * hypot(Float64(lambda1 * t_0), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(t_0 * Float64(-lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((phi1 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -1.02e-25)
		tmp = R * hypot((lambda1 * t_0), (phi1 - phi2));
	else
		tmp = R * hypot((t_0 * -lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -1.02e-25], N[(R * N[Sqrt[N[(lambda1 * t$95$0), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(t$95$0 * (-lambda2)), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.01999999999999998e-25

   1. Initial program 46.0%

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

      1. hypot-define 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in phi2 around 0 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in lambda1 around inf 71.7%

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

   if -1.01999999999999998e-25 < lambda1

   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. hypot-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi2 around 0 91.0%

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

      1. *-commutative 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in lambda1 around 0 82.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 82.6%

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

   10. Simplified 82.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.02 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.05e-19)
   (* R (hypot (* lambda1 (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.05e-19) {
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.05e-19) {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.05e-19:
		tmp = R * math.hypot((lambda1 * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.05e-19)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.05e-19)
		tmp = R * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.05e-19], 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[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.0499999999999999e-19

   1. Initial program 61.7%

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

      1. hypot-define 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in phi2 around 0 94.8%

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

      1. *-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in lambda1 around inf 86.2%

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

   if -1.0499999999999999e-19 < phi1

   1. Initial program 62.6%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi2 around 0 87.1%

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

      1. *-commutative 87.1%

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

   7. Simplified 87.1%

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

   8. Taylor expanded in phi1 around 0 50.8%

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

      1. unpow2 50.8%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 50.8%

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

      3. hypot-define 71.2%

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

   10. Simplified 71.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.5% 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 62.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in phi2 around 0 89.0%

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

   1. *-commutative 89.0%

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

7. Simplified 89.0%

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

8. Final simplification 89.0%

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

Alternative 6: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.75 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.75e+30)
   (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.75e+30) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.75e+30) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.75e+30:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.75e+30)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.75e+30)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.75e+30], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.75000000000000011e30

   1. Initial program 57.5%

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

      1. hypot-define 93.8%

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

   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 -inf 73.6%

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

      1. associate-*r* 73.6%

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

      2. mul-1-neg 73.6%

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

      3. associate-*r/ 73.6%

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

      4. mul-1-neg 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in R around -inf 77.3%

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

   if -1.75000000000000011e30 < phi1

   1. Initial program 63.7%

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

      1. hypot-define 97.2%

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

   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 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in phi1 around 0 51.2%

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

      1. unpow2 51.2%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 51.2%

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

      3. hypot-define 71.5%

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

   10. Simplified 71.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.75 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(\frac{\lambda_2}{\lambda_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.05e-142)
   (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
   (if (<= phi2 2.2e-47)
     (* R (* lambda1 (+ (/ lambda2 lambda1) -1.0)))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-142) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (phi2 <= 2.2e-47) {
		tmp = R * (lambda1 * ((lambda2 / lambda1) + -1.0));
	} 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 <= 1.05d-142) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else if (phi2 <= 2.2d-47) then
        tmp = r * (lambda1 * ((lambda2 / lambda1) + (-1.0d0)))
    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 <= 1.05e-142) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (phi2 <= 2.2e-47) {
		tmp = R * (lambda1 * ((lambda2 / lambda1) + -1.0));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.05e-142:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	elif phi2 <= 2.2e-47:
		tmp = R * (lambda1 * ((lambda2 / lambda1) + -1.0))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.05e-142)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	elseif (phi2 <= 2.2e-47)
		tmp = Float64(R * Float64(lambda1 * Float64(Float64(lambda2 / lambda1) + -1.0)));
	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 <= 1.05e-142)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	elseif (phi2 <= 2.2e-47)
		tmp = R * (lambda1 * ((lambda2 / lambda1) + -1.0));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.05e-142], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e-47], N[(R * N[(lambda1 * N[(N[(lambda2 / lambda1), $MachinePrecision] + -1.0), $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 < 1.05e-142

   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. Step-by-step derivation

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in phi1 around -inf 17.7%

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

      1. associate-*r* 17.7%

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

      2. mul-1-neg 17.7%

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

      3. associate-*r/ 17.7%

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

      4. mul-1-neg 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in R around -inf 19.0%

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

   if 1.05e-142 < phi2 < 2.20000000000000019e-47

   1. Initial program 64.7%

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

      1. hypot-define 99.9%

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

   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 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in lambda1 around -inf 37.9%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_1}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 37.9%

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

      2. mul-1-neg 37.9%

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

      3. unsub-neg 37.9%

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

      4. associate-/l* 37.8%

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

   10. Simplified 37.8%

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

   11. Taylor expanded in phi1 around 0 33.0%

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

       1. associate-*r* 33.0%

          \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \left(1 - \frac{\lambda_2}{\lambda_1}\right)\right)} \]

       2. neg-mul-1 33.0%

          \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \left(1 - \frac{\lambda_2}{\lambda_1}\right)\right) \]

   13. Simplified 33.0%

       \[\leadsto R \cdot \color{blue}{\left(\left(-\lambda_1\right) \cdot \left(1 - \frac{\lambda_2}{\lambda_1}\right)\right)} \]

   if 2.20000000000000019e-47 < phi2

   1. Initial program 63.3%

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

      1. hypot-define 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in phi2 around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. unsub-neg 68.1%

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

   7. Simplified 68.1%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(\frac{\lambda_2}{\lambda_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.6% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;R \leq 5.8 \cdot 10^{+251}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 5.8e+251)
   (- (* R phi2) (* R phi1))
   (* phi2 (- R (* phi1 (/ R phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 5.8e+251) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (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 (r <= 5.8d+251) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = phi2 * (r - (phi1 * (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 (R <= 5.8e+251) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if R <= 5.8e+251:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 5.8e+251)
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (R <= 5.8e+251)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 5.8e+251], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 5.7999999999999999e251

   1. Initial program 60.7%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in phi1 around -inf 32.0%

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

      1. associate-*r* 32.0%

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

      2. mul-1-neg 32.0%

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

      3. associate-*r/ 32.0%

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

      4. mul-1-neg 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in phi1 around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. mul-1-neg 33.8%

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

      3. unsub-neg 33.8%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 33.8%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   if 5.7999999999999999e251 < R

   1. Initial program 100.0%

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

      1. hypot-define 100.0%

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

   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 phi2 around inf 27.8%

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

      1. mul-1-neg 27.8%

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

      2. unsub-neg 27.8%

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

      3. *-commutative 27.8%

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

      4. associate-/l* 36.4%

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

   7. Simplified 36.4%

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

Alternative 9: 28.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;R \leq 5 \cdot 10^{+136}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 5e+136)
   (- (* R phi2) (* R phi1))
   (* R (* phi1 (+ (/ phi2 phi1) -1.0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 5e+136) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	}
	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 (r <= 5d+136) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 5e+136) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if R <= 5e+136:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 5e+136)
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (R <= 5e+136)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 5e+136], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 5.0000000000000002e136

   1. Initial program 55.2%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi1 around -inf 32.1%

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

      1. associate-*r* 32.1%

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

      2. mul-1-neg 32.1%

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

      3. associate-*r/ 32.1%

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

      4. mul-1-neg 32.1%

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

   7. Simplified 32.1%

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

   8. Taylor expanded in phi1 around 0 35.1%

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

      1. +-commutative 35.1%

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

      2. mul-1-neg 35.1%

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

      3. unsub-neg 35.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 35.1%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   if 5.0000000000000002e136 < R

   1. Initial program 100.0%

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

      1. hypot-define 100.0%

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

   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 -inf 34.8%

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

      1. associate-*r* 34.8%

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

      2. mul-1-neg 34.8%

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

      3. associate-*r/ 34.8%

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

      4. mul-1-neg 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in R around -inf 25.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 5 \cdot 10^{+136}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.4% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 8.5e+165) (- (* R phi2) (* R phi1)) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.5e+165) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * 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 <= 8.5d+165) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = r * 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 <= 8.5e+165) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 8.5e+165:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 8.5e+165)
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 8.5e+165)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 8.5e+165], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 8.5000000000000001e165

   1. Initial program 65.2%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi1 around -inf 32.9%

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

      1. associate-*r* 32.9%

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

      2. mul-1-neg 32.9%

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

      3. associate-*r/ 32.9%

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

      4. mul-1-neg 32.9%

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

   7. Simplified 32.9%

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

   8. Taylor expanded in phi1 around 0 34.0%

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

      1. +-commutative 34.0%

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

      2. mul-1-neg 34.0%

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

      3. unsub-neg 34.0%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 34.0%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   if 8.5000000000000001e165 < lambda2

   1. Initial program 37.6%

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

      1. hypot-define 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in phi2 around 0 78.8%

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

      1. *-commutative 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in lambda2 around inf 58.8%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \]

   9. Taylor expanded in phi1 around 0 63.0%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 28.2% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.95 \cdot 10^{+26}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.95e+26) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.95e+26) {
		tmp = R * -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 <= (-2.95d+26)) then
        tmp = r * -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 <= -2.95e+26) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.95e+26:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.95e+26)
		tmp = Float64(R * Float64(-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 <= -2.95e+26)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.95e+26], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.95000000000000015e26

   1. Initial program 57.3%

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

      1. hypot-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in phi1 around -inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. distribute-rgt-neg-in 76.5%

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

   7. Simplified 76.5%

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

   if -2.95000000000000015e26 < phi1

   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. hypot-define 97.2%

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

   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 inf 23.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 25.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7e-21) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7e-21) {
		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 <= 7d-21) 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 <= 7e-21) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7e-21:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7e-21)
		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 <= 7e-21)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7e-21], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.0000000000000007e-21

   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. Step-by-step derivation

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi2 around 0 90.0%

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

      1. *-commutative 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in lambda2 around inf 14.1%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \]

   9. Taylor expanded in phi1 around 0 11.5%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]

   if 7.0000000000000007e-21 < phi2

   1. Initial program 64.3%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi2 around inf 68.6%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 13.7% 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 62.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in phi2 around 0 89.0%

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

   1. *-commutative 89.0%

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

7. Simplified 89.0%

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

8. Taylor expanded in lambda2 around inf 13.5%

   \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \]

9. Taylor expanded in phi1 around 0 11.4%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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))))))