Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 63.3%

Time: 3.0s

Alternatives: 9

Speedup: 1.0×

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

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 59.6% 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: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (if (<= (- lambda1 lambda2) 2.1e+139)
     (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= 2.1e+139) {
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * (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) :: t_0
    real(8) :: tmp
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    if ((lambda1 - lambda2) <= 2.1d+139) then
        tmp = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
    else
        tmp = r * (lambda1 - lambda2)
    end if
    code = tmp
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));
	double tmp;
	if ((lambda1 - lambda2) <= 2.1e+139) {
		tmp = R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * (lambda1 - lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	tmp = 0
	if (lambda1 - lambda2) <= 2.1e+139:
		tmp = R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
	else:
		tmp = R * (lambda1 - lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= 2.1e+139)
		tmp = Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * Float64(lambda1 - lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = 0.0;
	if ((lambda1 - lambda2) <= 2.1e+139)
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	else
		tmp = R * (lambda1 - lambda2);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 2.1e+139], 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], N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < 2.0999999999999999e139

   1. Initial program 67.3%

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

   if 2.0999999999999999e139 < (-.f64 lambda1 lambda2)

   1. Initial program 39.0%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 37.2%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 59.4%

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

Alternative 2: 44.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \left(t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 4 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (if (<= (- lambda1 lambda2) -1.1e-12)
     (* R (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2))))
     (if (<= (- lambda1 lambda2) 4e-50)
       (* R (+ phi1 phi2))
       (* R (- lambda1 lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= -1.1e-12) {
		tmp = R * ((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (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) :: t_0
    real(8) :: tmp
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    if ((lambda1 - lambda2) <= (-1.1d-12)) then
        tmp = r * ((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)))
    else if ((lambda1 - lambda2) <= 4d-50) then
        tmp = r * (phi1 + phi2)
    else
        tmp = r * (lambda1 - lambda2)
    end if
    code = tmp
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));
	double tmp;
	if ((lambda1 - lambda2) <= -1.1e-12) {
		tmp = R * ((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (lambda1 - lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	tmp = 0
	if (lambda1 - lambda2) <= -1.1e-12:
		tmp = R * ((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)))
	elif (lambda1 - lambda2) <= 4e-50:
		tmp = R * (phi1 + phi2)
	else:
		tmp = R * (lambda1 - lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1.1e-12)
		tmp = Float64(R * Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))));
	elseif (Float64(lambda1 - lambda2) <= 4e-50)
		tmp = Float64(R * Float64(phi1 + phi2));
	else
		tmp = Float64(R * Float64(lambda1 - lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1.1e-12)
		tmp = R * ((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)));
	elseif ((lambda1 - lambda2) <= 4e-50)
		tmp = R * (phi1 + phi2);
	else
		tmp = R * (lambda1 - lambda2);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1.1e-12], N[(R * N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 4e-50], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1.09999999999999996e-12

   1. Initial program 58.7%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 43.7%

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

   if -1.09999999999999996e-12 < (-.f64 lambda1 lambda2) < 4.00000000000000003e-50

   1. Initial program 68.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 50.5%

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

   if 4.00000000000000003e-50 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 50.9%

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

Alternative 3: 45.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.35 \cdot 10^{+124}:\\ \;\;\;\;R \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 4 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (if (<= (- lambda1 lambda2) -1.35e+124)
     (* R (* t_0 t_0))
     (if (<= (- lambda1 lambda2) 4e-50)
       (* R (+ phi1 phi2))
       (* R (- lambda1 lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= -1.35e+124) {
		tmp = R * (t_0 * t_0);
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (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) :: t_0
    real(8) :: tmp
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    if ((lambda1 - lambda2) <= (-1.35d+124)) then
        tmp = r * (t_0 * t_0)
    else if ((lambda1 - lambda2) <= 4d-50) then
        tmp = r * (phi1 + phi2)
    else
        tmp = r * (lambda1 - lambda2)
    end if
    code = tmp
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));
	double tmp;
	if ((lambda1 - lambda2) <= -1.35e+124) {
		tmp = R * (t_0 * t_0);
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (lambda1 - lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	tmp = 0
	if (lambda1 - lambda2) <= -1.35e+124:
		tmp = R * (t_0 * t_0)
	elif (lambda1 - lambda2) <= 4e-50:
		tmp = R * (phi1 + phi2)
	else:
		tmp = R * (lambda1 - lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1.35e+124)
		tmp = Float64(R * Float64(t_0 * t_0));
	elseif (Float64(lambda1 - lambda2) <= 4e-50)
		tmp = Float64(R * Float64(phi1 + phi2));
	else
		tmp = Float64(R * Float64(lambda1 - lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1.35e+124)
		tmp = R * (t_0 * t_0);
	elseif ((lambda1 - lambda2) <= 4e-50)
		tmp = R * (phi1 + phi2);
	else
		tmp = R * (lambda1 - lambda2);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1.35e+124], N[(R * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 4e-50], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1.34999999999999989e124

   1. Initial program 53.4%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \]

   8. Applied rewrites 47.7%

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

   if -1.34999999999999989e124 < (-.f64 lambda1 lambda2) < 4.00000000000000003e-50

   1. Initial program 69.3%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 43.6%

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

   if 4.00000000000000003e-50 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 50.9%

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

Alternative 4: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\phi_1 + \phi_2}{2}\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \left(R \cdot t\_0\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (/ (+ phi1 phi2) 2.0)))
   (if (<= phi2 -7.5e+54)
     (* R (* R t_0))
     (if (<= phi2 3.8e-45)
       (* R (* (- lambda1 lambda2) (cos t_0)))
       (* R (+ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 + phi2) / 2.0;
	double tmp;
	if (phi2 <= -7.5e+54) {
		tmp = R * (R * t_0);
	} else if (phi2 <= 3.8e-45) {
		tmp = R * ((lambda1 - lambda2) * cos(t_0));
	} else {
		tmp = R * (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) :: t_0
    real(8) :: tmp
    t_0 = (phi1 + phi2) / 2.0d0
    if (phi2 <= (-7.5d+54)) then
        tmp = r * (r * t_0)
    else if (phi2 <= 3.8d-45) then
        tmp = r * ((lambda1 - lambda2) * cos(t_0))
    else
        tmp = r * (phi1 + phi2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 + phi2) / 2.0;
	double tmp;
	if (phi2 <= -7.5e+54) {
		tmp = R * (R * t_0);
	} else if (phi2 <= 3.8e-45) {
		tmp = R * ((lambda1 - lambda2) * Math.cos(t_0));
	} else {
		tmp = R * (phi1 + phi2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (phi1 + phi2) / 2.0
	tmp = 0
	if phi2 <= -7.5e+54:
		tmp = R * (R * t_0)
	elif phi2 <= 3.8e-45:
		tmp = R * ((lambda1 - lambda2) * math.cos(t_0))
	else:
		tmp = R * (phi1 + phi2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 + phi2) / 2.0)
	tmp = 0.0
	if (phi2 <= -7.5e+54)
		tmp = Float64(R * Float64(R * t_0));
	elseif (phi2 <= 3.8e-45)
		tmp = Float64(R * Float64(Float64(lambda1 - lambda2) * cos(t_0)));
	else
		tmp = Float64(R * Float64(phi1 + phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (phi1 + phi2) / 2.0;
	tmp = 0.0;
	if (phi2 <= -7.5e+54)
		tmp = R * (R * t_0);
	elseif (phi2 <= 3.8e-45)
		tmp = R * ((lambda1 - lambda2) * cos(t_0));
	else
		tmp = R * (phi1 + phi2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[phi2, -7.5e+54], N[(R * N[(R * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.8e-45], N[(R * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.50000000000000042e54

   1. Initial program 50.7%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 39.8%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 25.2%

       \[\leadsto R \cdot \color{blue}{\left(R \cdot \frac{\phi_1 + \phi_2}{2}\right)} \]

   if -7.50000000000000042e54 < phi2 < 3.79999999999999997e-45

   1. Initial program 68.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 41.1%

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

   if 3.79999999999999997e-45 < phi2

   1. Initial program 51.0%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 58.4%

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

Alternative 5: 39.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.8 \cdot 10^{+154}:\\ \;\;\;\;R \cdot \left(R \cdot \frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 4 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -3.8e+154)
   (* R (* R (/ (+ phi1 phi2) 2.0)))
   (if (<= (- lambda1 lambda2) 4e-50)
     (* R (+ phi1 phi2))
     (* R (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -3.8e+154) {
		tmp = R * (R * ((phi1 + phi2) / 2.0));
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (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 ((lambda1 - lambda2) <= (-3.8d+154)) then
        tmp = r * (r * ((phi1 + phi2) / 2.0d0))
    else if ((lambda1 - lambda2) <= 4d-50) then
        tmp = r * (phi1 + phi2)
    else
        tmp = r * (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 ((lambda1 - lambda2) <= -3.8e+154) {
		tmp = R * (R * ((phi1 + phi2) / 2.0));
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (lambda1 - lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -3.8e+154:
		tmp = R * (R * ((phi1 + phi2) / 2.0))
	elif (lambda1 - lambda2) <= 4e-50:
		tmp = R * (phi1 + phi2)
	else:
		tmp = R * (lambda1 - lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -3.8e+154)
		tmp = Float64(R * Float64(R * Float64(Float64(phi1 + phi2) / 2.0)));
	elseif (Float64(lambda1 - lambda2) <= 4e-50)
		tmp = Float64(R * Float64(phi1 + phi2));
	else
		tmp = Float64(R * Float64(lambda1 - lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -3.8e+154)
		tmp = R * (R * ((phi1 + phi2) / 2.0));
	elseif ((lambda1 - lambda2) <= 4e-50)
		tmp = R * (phi1 + phi2);
	else
		tmp = R * (lambda1 - lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -3.8e+154], N[(R * N[(R * N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 4e-50], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -3.7999999999999998e154

   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. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 28.2%

       \[\leadsto R \cdot \color{blue}{\left(R \cdot \frac{\phi_1 + \phi_2}{2}\right)} \]

   if -3.7999999999999998e154 < (-.f64 lambda1 lambda2) < 4.00000000000000003e-50

   1. Initial program 69.5%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 39.7%

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

   if 4.00000000000000003e-50 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 50.9%

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

Alternative 6: 41.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -7.2 \cdot 10^{+192}:\\ \;\;\;\;R \cdot t\_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 4 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1 lambda2))))
   (if (<= (- lambda1 lambda2) -7.2e+192)
     (* R t_0)
     (if (<= (- lambda1 lambda2) 4e-50) (* R (+ phi1 phi2)) t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * (lambda1 - lambda2);
	double tmp;
	if ((lambda1 - lambda2) <= -7.2e+192) {
		tmp = R * t_0;
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = r * (lambda1 - lambda2)
    if ((lambda1 - lambda2) <= (-7.2d+192)) then
        tmp = r * t_0
    else if ((lambda1 - lambda2) <= 4d-50) then
        tmp = r * (phi1 + phi2)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * (lambda1 - lambda2);
	double tmp;
	if ((lambda1 - lambda2) <= -7.2e+192) {
		tmp = R * t_0;
	} else if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * (lambda1 - lambda2)
	tmp = 0
	if (lambda1 - lambda2) <= -7.2e+192:
		tmp = R * t_0
	elif (lambda1 - lambda2) <= 4e-50:
		tmp = R * (phi1 + phi2)
	else:
		tmp = t_0
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(lambda1 - lambda2))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -7.2e+192)
		tmp = Float64(R * t_0);
	elseif (Float64(lambda1 - lambda2) <= 4e-50)
		tmp = Float64(R * Float64(phi1 + phi2));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * (lambda1 - lambda2);
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -7.2e+192)
		tmp = R * t_0;
	elseif ((lambda1 - lambda2) <= 4e-50)
		tmp = R * (phi1 + phi2);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -7.2e+192], N[(R * t$95$0), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 4e-50], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -7.2000000000000004e192

   1. Initial program 43.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 20.9%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \]

   8. Applied rewrites 43.6%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   11. Applied rewrites 26.1%

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

   if -7.2000000000000004e192 < (-.f64 lambda1 lambda2) < 4.00000000000000003e-50

   1. Initial program 70.1%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 39.7%

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

   if 4.00000000000000003e-50 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 50.9%

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

Alternative 7: 39.7% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq 4 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) 4e-50)
   (* R (+ phi1 phi2))
   (* R (- lambda1 lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (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 ((lambda1 - lambda2) <= 4d-50) then
        tmp = r * (phi1 + phi2)
    else
        tmp = r * (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 ((lambda1 - lambda2) <= 4e-50) {
		tmp = R * (phi1 + phi2);
	} else {
		tmp = R * (lambda1 - lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= 4e-50:
		tmp = R * (phi1 + phi2)
	else:
		tmp = R * (lambda1 - lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= 4e-50)
		tmp = Float64(R * Float64(phi1 + phi2));
	else
		tmp = Float64(R * Float64(lambda1 - lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= 4e-50)
		tmp = R * (phi1 + phi2);
	else
		tmp = R * (lambda1 - lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], 4e-50], N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < 4.00000000000000003e-50

   1. Initial program 62.1%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 13.6%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

   8. Applied rewrites 30.7%

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

   if 4.00000000000000003e-50 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \color{blue}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

   11. Applied rewrites 50.9%

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

Alternative 8: 29.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\phi_1 + \phi_2\right) \end{array} \]

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (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
    code = r * (phi1 + phi2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in lambda1 around 0

   \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

5. Applied rewrites 27.1%

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

6. Taylor expanded in lambda1 around 0

   \[\leadsto R \cdot \color{blue}{\left(\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)\right)} \]

8. Applied rewrites 28.0%

   \[\leadsto R \cdot \color{blue}{\left(\phi_1 + \phi_2\right)} \]
9. Add Preprocessing

Alternative 9: 3.3% accurate, 69.8× speedup

Math

\[\begin{array}{l} \\ \lambda_1 - \lambda_2 \end{array} \]

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 - 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 = lambda1 - lambda2
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return lambda1 - lambda2

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 - lambda2), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in lambda1 around 0

   \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

5. Applied rewrites 27.1%

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

6. Taylor expanded in lambda1 around 0

   \[\leadsto \color{blue}{R \cdot \sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} + \lambda_1 \cdot \left(-1 \cdot \left(\left(R \cdot \left(\lambda_2 \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right) + \frac{1}{2} \cdot \left(\left(R \cdot \left(\lambda_1 \cdot \left({\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} - \frac{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{4}}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}}\right)\right)} \]

8. Applied rewrites 3.3%

   \[\leadsto \color{blue}{\lambda_1 - \lambda_2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  :pre (TRUE)
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))