Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.4% → 99.9%

Time: 20.8s

Alternatives: 17

Speedup: 3.0×

• 31.0% 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: 59.4% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (-
     (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
     (expm1 (log1p (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))))
    (- lambda1 lambda2))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - expm1(log1p((sin((0.5 * phi1)) * sin((phi2 * 0.5)))))) * (lambda1 - lambda2)), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - Math.expm1(Math.log1p((Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5)))))) * (lambda1 - lambda2)), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - math.expm1(math.log1p((math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5)))))) * (lambda1 - lambda2)), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - expm1(log1p(Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))))) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))
end

Wolfram

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

Derivation

1. Initial program 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

   1. add-cube-cbrt 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \]

   2. pow3 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]

   3. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}}\right)}^{3}, \phi_1 - \phi_2\right) \]

   4. div-inv 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. metadata-eval 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

5. Applied egg-rr 94.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   2. +-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   3. distribute-rgt-in 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\phi_2 \cdot 0.5 + \phi_1 \cdot 0.5\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   4. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\color{blue}{0.5 \cdot \phi_2} + \phi_1 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. cos-sum 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   6. *-commutative 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   7. *-commutative 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. rem-cube-cbrt 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. expm1-log1p-u 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \]

12. Final simplification 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \]

Alternative 2: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1.14 \cdot 10^{-57}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t_1 - t_0\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_0 - t_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
        (t_1 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (if (<= lambda1 -1.14e-57)
     (* R (hypot (* lambda1 (- t_1 t_0)) (- phi1 phi2)))
     (* R (hypot (* lambda2 (- t_0 t_1)) (- phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.14e-57) {
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5));
	double t_1 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.14e-57) {
		tmp = R * Math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))
	t_1 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -1.14e-57:
		tmp = R * math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	tmp = 0.0
	if (lambda1 <= -1.14e-57)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(t_1 - t_0)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(t_0 - t_1)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -1.14e-57)
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.14e-57], N[(R * N[Sqrt[N[(lambda1 * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.14000000000000006e-57

   1. Initial program 54.8%

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

      1. hypot-def 94.7%

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

   3. Simplified 94.7%

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

      1. add-cube-cbrt 94.3%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \]

      2. pow3 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]

      3. *-commutative 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}}\right)}^{3}, \phi_1 - \phi_2\right) \]

      4. div-inv 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      5. metadata-eval 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 94.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      2. +-commutative 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      3. distribute-rgt-in 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\phi_2 \cdot 0.5 + \phi_1 \cdot 0.5\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      4. *-commutative 94.4%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\color{blue}{0.5 \cdot \phi_2} + \phi_1 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      5. cos-sum 99.5%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      6. *-commutative 99.5%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      7. *-commutative 99.5%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 99.5%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda2 around 0 88.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{1}^{0.3333333333333333} \cdot \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1\right)}, \phi_1 - \phi_2\right) \]
   9. Step-by-step derivation

      1. pow-base-1 88.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{1} \cdot \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1\right), \phi_1 - \phi_2\right) \]

      2. *-lft-identity 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

   10. Simplified 88.6%

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

   if -1.14000000000000006e-57 < lambda1

   1. Initial program 60.4%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

      1. add-cube-cbrt 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \]

      2. pow3 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]

      3. *-commutative 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}}\right)}^{3}, \phi_1 - \phi_2\right) \]

      4. div-inv 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      5. metadata-eval 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 95.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      2. +-commutative 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      3. distribute-rgt-in 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\phi_2 \cdot 0.5 + \phi_1 \cdot 0.5\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      4. *-commutative 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\color{blue}{0.5 \cdot \phi_2} + \phi_1 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      5. cos-sum 99.3%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      6. *-commutative 99.3%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

      7. *-commutative 99.3%

         \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 99.3%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in lambda1 around 0 86.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)}, \phi_1 - \phi_2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg 86.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-{1}^{0.3333333333333333} \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. pow-base-1 86.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(-\color{blue}{1} \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right), \phi_1 - \phi_2\right) \]

      3. *-lft-identity 86.5%

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

      4. *-commutative 86.5%

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

      5. distribute-rgt-neg-in 86.5%

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

      6. *-commutative 86.5%

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

   10. Simplified 86.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.14 \cdot 10^{-57}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

   1. add-cube-cbrt 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \]

   2. pow3 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]

   3. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}}\right)}^{3}, \phi_1 - \phi_2\right) \]

   4. div-inv 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. metadata-eval 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

5. Applied egg-rr 94.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   2. +-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   3. distribute-rgt-in 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \color{blue}{\left(\phi_2 \cdot 0.5 + \phi_1 \cdot 0.5\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   4. *-commutative 94.8%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\cos \left(\color{blue}{0.5 \cdot \phi_2} + \phi_1 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   5. cos-sum 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   6. *-commutative 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

   7. *-commutative 99.4%

      \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto R \cdot \mathsf{hypot}\left({\left(\sqrt[3]{\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)}\right)}^{3}, \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. rem-cube-cbrt 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 4: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_0 - \lambda_2 \cdot t_0, \phi_1 - \phi_2\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 (+ phi2 phi1)))))
   (* R (hypot (- (* lambda1 t_0) (* lambda2 t_0)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * (phi2 + phi1)));
	return R * hypot(((lambda1 * t_0) - (lambda2 * t_0)), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * (phi2 + phi1)));
	return R * Math.hypot(((lambda1 * t_0) - (lambda2 * t_0)), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * (phi2 + phi1)))
	return R * math.hypot(((lambda1 * t_0) - (lambda2 * t_0)), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	return Float64(R * hypot(Float64(Float64(lambda1 * t_0) - Float64(lambda2 * t_0)), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * (phi2 + phi1)));
	tmp = R * hypot(((lambda1 * t_0) - (lambda2 * t_0)), (phi1 - phi2));
end

Wolfram

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

Derivation

1. Initial program 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

   1. *-commutative 95.2%

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

   2. sub-neg 95.2%

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

   3. distribute-lft-in 95.2%

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

   4. div-inv 95.2%

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

   5. metadata-eval 95.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \lambda_1 + \cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right) \]

   6. div-inv 95.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \lambda_1 + \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right) \]

   7. metadata-eval 95.2%

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

5. Applied egg-rr 95.2%

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

6. Final simplification 95.2%

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

Alternative 5: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1e-142)
   (* R (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))))
   (* 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 <= 1e-142) {
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} 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 <= 1e-142) {
		tmp = R * Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} 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 <= 1e-142:
		tmp = R * math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2)))
	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 <= 1e-142)
		tmp = Float64(R * hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))));
	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 <= 1e-142)
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	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, 1e-142], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $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 < 1e-142

   1. Initial program 61.9%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in phi2 around 0 52.7%

      \[\leadsto \color{blue}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \cdot R} \]
   5. Step-by-step derivation

      1. *-commutative 52.7%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]

      2. +-commutative 52.7%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      3. unpow2 52.7%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 52.7%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 52.7%

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

      6. unswap-sqr 52.7%

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

      7. hypot-def 76.0%

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

   6. Simplified 76.0%

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

   if 1e-142 < phi2

   1. Initial program 52.9%

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

      1. hypot-def 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in phi1 around 0 92.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 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[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Final simplification 95.2%

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

Alternative 7: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.25e-142)
   (* R (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))))
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-142) {
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} else {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-142) {
		tmp = R * Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.25e-142:
		tmp = R * math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2)))
	else:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.25e-142)
		tmp = Float64(R * hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))));
	else
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.25e-142)
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	else
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.25e-142], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.2500000000000001e-142

   1. Initial program 61.9%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in phi2 around 0 52.7%

      \[\leadsto \color{blue}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \cdot R} \]
   5. Step-by-step derivation

      1. *-commutative 52.7%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]

      2. +-commutative 52.7%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      3. unpow2 52.7%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 52.7%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 52.7%

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

      6. unswap-sqr 52.7%

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

      7. hypot-def 76.0%

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

   6. Simplified 76.0%

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

   if 1.2500000000000001e-142 < phi2

   1. Initial program 52.9%

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

      1. hypot-def 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in phi1 around 0 92.8%

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

   5. Taylor expanded in phi2 around 0 86.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 8: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.4e+14)
   (* R (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.4e+14) {
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.4e+14) {
		tmp = R * Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2)));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.4e+14:
		tmp = R * math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2)))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.4e+14)
		tmp = Float64(R * hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.4e+14)
		tmp = R * hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2)));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.4e+14], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.4e14

   1. Initial program 62.8%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in phi2 around 0 53.9%

      \[\leadsto \color{blue}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \cdot R} \]
   5. Step-by-step derivation

      1. *-commutative 53.9%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]

      2. +-commutative 53.9%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      3. unpow2 53.9%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 53.9%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 53.9%

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

      6. unswap-sqr 53.9%

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

      7. hypot-def 77.3%

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

   6. Simplified 77.3%

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

   if 5.4e14 < phi2

   1. Initial program 48.5%

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

      1. hypot-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in phi1 around 0 46.0%

      \[\leadsto \color{blue}{\sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   5. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 46.0%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 46.0%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 46.0%

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

      5. unswap-sqr 46.0%

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

      6. hypot-def 78.3%

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

   6. Simplified 78.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 74.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.5 \cdot 10^{+167}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 8.5e+167)
   (* R (hypot lambda1 (- phi1 phi2)))
   (* R (- lambda2 lambda1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.5e+167) {
		tmp = R * hypot(lambda1, (phi1 - phi2));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.5e+167) {
		tmp = R * Math.hypot(lambda1, (phi1 - phi2));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 8.5e+167:
		tmp = R * math.hypot(lambda1, (phi1 - phi2))
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 8.5e+167)
		tmp = Float64(R * hypot(lambda1, Float64(phi1 - phi2)));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 8.5e+167)
		tmp = R * hypot(lambda1, (phi1 - phi2));
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 8.5e+167], N[(R * N[Sqrt[lambda1 ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 8.50000000000000007e167

   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-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in phi1 around 0 91.5%

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

   5. Taylor expanded in phi2 around 0 88.2%

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

   6. Taylor expanded in lambda2 around 0 54.5%

      \[\leadsto \color{blue}{\sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 54.5%

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

      2. unpow2 54.5%

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

      3. unpow2 54.5%

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

      4. hypot-def 81.0%

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

   8. Simplified 81.0%

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

   if 8.50000000000000007e167 < lambda2

   1. Initial program 35.6%

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

      1. hypot-def 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in phi1 around 0 76.8%

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

   5. Taylor expanded in phi2 around 0 73.2%

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

   6. Taylor expanded in lambda1 around -inf 58.9%

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

      1. *-commutative 58.9%

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

      2. neg-mul-1 58.9%

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

      3. sub-neg 58.9%

         \[\leadsto \color{blue}{\lambda_2 \cdot R - \lambda_1 \cdot R} \]

      4. distribute-rgt-out-- 62.4%

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

   8. Simplified 62.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.5 \cdot 10^{+167}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 10: 73.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.16e+35)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot lambda1 (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.16e+35) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(lambda1, (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.16e+35) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(lambda1, (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.16e+35:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(lambda1, (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.16e+35)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(lambda1, Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.16e+35)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(lambda1, (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.16e+35], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[lambda1 ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.1600000000000001e35

   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-def 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in phi1 around 0 88.0%

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

   5. Taylor expanded in phi2 around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. hypot-def 69.9%

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

   7. Simplified 69.9%

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

   if 1.1600000000000001e35 < phi2

   1. Initial program 48.3%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in phi1 around 0 95.2%

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

   5. Taylor expanded in phi2 around 0 88.0%

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

   6. Taylor expanded in lambda2 around 0 45.8%

      \[\leadsto \color{blue}{\sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 45.8%

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

      2. unpow2 45.8%

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

      3. unpow2 45.8%

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

      4. hypot-def 86.1%

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

   8. Simplified 86.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 11: 85.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in phi1 around 0 89.9%

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

5. Taylor expanded in phi2 around 0 86.6%

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

6. Final simplification 86.6%

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

Alternative 12: 28.6% accurate, 20.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -8.6 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.12 \cdot 10^{-234}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- phi1))))
   (if (<= phi2 -8.6e-169)
     t_0
     (if (<= phi2 -1e-292)
       (* R lambda2)
       (if (<= phi2 1.55e-298)
         t_0
         (if (<= phi2 1.12e-234)
           (* R (- lambda1))
           (if (<= phi2 5e-225)
             (* R lambda2)
             (if (<= phi2 3e+31) t_0 (* R phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -phi1;
	double tmp;
	if (phi2 <= -8.6e-169) {
		tmp = t_0;
	} else if (phi2 <= -1e-292) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.55e-298) {
		tmp = t_0;
	} else if (phi2 <= 1.12e-234) {
		tmp = R * -lambda1;
	} else if (phi2 <= 5e-225) {
		tmp = R * lambda2;
	} else if (phi2 <= 3e+31) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = r * -phi1
    if (phi2 <= (-8.6d-169)) then
        tmp = t_0
    else if (phi2 <= (-1d-292)) then
        tmp = r * lambda2
    else if (phi2 <= 1.55d-298) then
        tmp = t_0
    else if (phi2 <= 1.12d-234) then
        tmp = r * -lambda1
    else if (phi2 <= 5d-225) then
        tmp = r * lambda2
    else if (phi2 <= 3d+31) then
        tmp = t_0
    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 t_0 = R * -phi1;
	double tmp;
	if (phi2 <= -8.6e-169) {
		tmp = t_0;
	} else if (phi2 <= -1e-292) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.55e-298) {
		tmp = t_0;
	} else if (phi2 <= 1.12e-234) {
		tmp = R * -lambda1;
	} else if (phi2 <= 5e-225) {
		tmp = R * lambda2;
	} else if (phi2 <= 3e+31) {
		tmp = t_0;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -phi1
	tmp = 0
	if phi2 <= -8.6e-169:
		tmp = t_0
	elif phi2 <= -1e-292:
		tmp = R * lambda2
	elif phi2 <= 1.55e-298:
		tmp = t_0
	elif phi2 <= 1.12e-234:
		tmp = R * -lambda1
	elif phi2 <= 5e-225:
		tmp = R * lambda2
	elif phi2 <= 3e+31:
		tmp = t_0
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-phi1))
	tmp = 0.0
	if (phi2 <= -8.6e-169)
		tmp = t_0;
	elseif (phi2 <= -1e-292)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 1.55e-298)
		tmp = t_0;
	elseif (phi2 <= 1.12e-234)
		tmp = Float64(R * Float64(-lambda1));
	elseif (phi2 <= 5e-225)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 3e+31)
		tmp = t_0;
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -phi1;
	tmp = 0.0;
	if (phi2 <= -8.6e-169)
		tmp = t_0;
	elseif (phi2 <= -1e-292)
		tmp = R * lambda2;
	elseif (phi2 <= 1.55e-298)
		tmp = t_0;
	elseif (phi2 <= 1.12e-234)
		tmp = R * -lambda1;
	elseif (phi2 <= 5e-225)
		tmp = R * lambda2;
	elseif (phi2 <= 3e+31)
		tmp = t_0;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-phi1)), $MachinePrecision]}, If[LessEqual[phi2, -8.6e-169], t$95$0, If[LessEqual[phi2, -1e-292], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 1.55e-298], t$95$0, If[LessEqual[phi2, 1.12e-234], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[phi2, 5e-225], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 3e+31], t$95$0, N[(R * phi2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -8.59999999999999967e-169 or -1.0000000000000001e-292 < phi2 < 1.5500000000000001e-298 or 5.0000000000000001e-225 < phi2 < 2.99999999999999989e31

   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-def 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in phi1 around -inf 19.5%

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

      1. associate-*r* 19.5%

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

      2. mul-1-neg 19.5%

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

   6. Simplified 19.5%

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

   if -8.59999999999999967e-169 < phi2 < -1.0000000000000001e-292 or 1.11999999999999998e-234 < phi2 < 5.0000000000000001e-225

   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. Step-by-step derivation

      1. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 80.4%

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

   5. Taylor expanded in phi2 around 0 80.4%

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

   6. Taylor expanded in lambda2 around inf 21.0%

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

   if 1.5500000000000001e-298 < phi2 < 1.11999999999999998e-234

   1. Initial program 68.1%

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

      1. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in phi1 around 0 80.4%

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

   5. Taylor expanded in phi2 around 0 80.4%

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

   6. Taylor expanded in lambda1 around -inf 27.0%

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

      1. *-commutative 27.0%

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

      2. neg-mul-1 27.0%

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

   8. Simplified 27.0%

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

   if 2.99999999999999989e31 < phi2

   1. Initial program 48.3%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in phi2 around inf 65.4%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.6 \cdot 10^{-169}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{-298}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.12 \cdot 10^{-234}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{+31}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13: 25.5% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.4 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.4e-162)
   (* R lambda2)
   (if (<= phi2 2.6e-47)
     (* R (- lambda1))
     (if (<= phi2 1.16e+35) (* R lambda2) (* R phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.4e-162) {
		tmp = R * lambda2;
	} else if (phi2 <= 2.6e-47) {
		tmp = R * -lambda1;
	} else if (phi2 <= 1.16e+35) {
		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 <= 7.4d-162) then
        tmp = r * lambda2
    else if (phi2 <= 2.6d-47) then
        tmp = r * -lambda1
    else if (phi2 <= 1.16d+35) 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 <= 7.4e-162) {
		tmp = R * lambda2;
	} else if (phi2 <= 2.6e-47) {
		tmp = R * -lambda1;
	} else if (phi2 <= 1.16e+35) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.4e-162:
		tmp = R * lambda2
	elif phi2 <= 2.6e-47:
		tmp = R * -lambda1
	elif phi2 <= 1.16e+35:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.4e-162)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 2.6e-47)
		tmp = Float64(R * Float64(-lambda1));
	elseif (phi2 <= 1.16e+35)
		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 <= 7.4e-162)
		tmp = R * lambda2;
	elseif (phi2 <= 2.6e-47)
		tmp = R * -lambda1;
	elseif (phi2 <= 1.16e+35)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.4e-162], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 2.6e-47], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[phi2, 1.16e+35], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 7.4000000000000003e-162 or 2.6e-47 < phi2 < 1.1600000000000001e35

   1. Initial program 61.3%

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

      1. hypot-def 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in phi1 around 0 87.3%

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

   5. Taylor expanded in phi2 around 0 85.2%

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

   6. Taylor expanded in lambda2 around inf 17.8%

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

   if 7.4000000000000003e-162 < phi2 < 2.6e-47

   1. Initial program 75.9%

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

      1. hypot-def 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. Taylor expanded in phi1 around 0 95.6%

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

   5. Taylor expanded in phi2 around 0 95.6%

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

   6. Taylor expanded in lambda1 around -inf 18.1%

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

      1. *-commutative 18.1%

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

      2. neg-mul-1 18.1%

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

   8. Simplified 18.1%

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

   if 1.1600000000000001e35 < phi2

   1. Initial program 48.3%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in phi2 around inf 65.4%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.4 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14: 32.2% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-152}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -7.5e-152)
   (* R (- phi1))
   (if (<= phi2 1.1e+37) (* R (- lambda2 lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -7.5e-152) {
		tmp = R * -phi1;
	} else if (phi2 <= 1.1e+37) {
		tmp = R * (lambda2 - lambda1);
	} 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 <= (-7.5d-152)) then
        tmp = r * -phi1
    else if (phi2 <= 1.1d+37) then
        tmp = r * (lambda2 - lambda1)
    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 <= -7.5e-152) {
		tmp = R * -phi1;
	} else if (phi2 <= 1.1e+37) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -7.5e-152:
		tmp = R * -phi1
	elif phi2 <= 1.1e+37:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -7.5e-152)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi2 <= 1.1e+37)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -7.5e-152)
		tmp = R * -phi1;
	elseif (phi2 <= 1.1e+37)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -7.5e-152], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi2, 1.1e+37], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.5e-152

   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-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in phi1 around -inf 15.6%

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

      1. associate-*r* 15.6%

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

      2. mul-1-neg 15.6%

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

   6. Simplified 15.6%

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

   if -7.5e-152 < phi2 < 1.1e37

   1. Initial program 61.8%

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

      1. hypot-def 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in phi1 around 0 84.8%

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

   5. Taylor expanded in phi2 around 0 84.0%

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

   6. Taylor expanded in lambda1 around -inf 33.2%

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

      1. *-commutative 33.2%

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

      2. neg-mul-1 33.2%

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

      3. sub-neg 33.2%

         \[\leadsto \color{blue}{\lambda_2 \cdot R - \lambda_1 \cdot R} \]

      4. distribute-rgt-out-- 33.2%

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

   8. Simplified 33.2%

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

   if 1.1e37 < phi2

   1. Initial program 48.3%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in phi2 around inf 65.4%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-152}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15: 33.9% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-157}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -9e-157)
   (* R (- phi1))
   (if (<= phi2 9.5e-38) (* R (- lambda2 lambda1)) (* R (- phi2 phi1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9e-157) {
		tmp = R * -phi1;
	} else if (phi2 <= 9.5e-38) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= (-9d-157)) then
        tmp = r * -phi1
    else if (phi2 <= 9.5d-38) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * (phi2 - phi1)
    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 <= -9e-157) {
		tmp = R * -phi1;
	} else if (phi2 <= 9.5e-38) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -9e-157:
		tmp = R * -phi1
	elif phi2 <= 9.5e-38:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -9e-157)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi2 <= 9.5e-38)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -9e-157)
		tmp = R * -phi1;
	elseif (phi2 <= 9.5e-38)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -9e-157], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi2, 9.5e-38], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -8.99999999999999997e-157

   1. Initial program 62.7%

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

      1. hypot-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in phi1 around -inf 15.4%

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

      1. associate-*r* 15.4%

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

      2. mul-1-neg 15.4%

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

   6. Simplified 15.4%

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

   if -8.99999999999999997e-157 < phi2 < 9.5000000000000009e-38

   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-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 86.0%

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

   5. Taylor expanded in phi2 around 0 86.0%

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

   6. Taylor expanded in lambda1 around -inf 33.2%

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

      1. *-commutative 33.2%

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

      2. neg-mul-1 33.2%

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

      3. sub-neg 33.2%

         \[\leadsto \color{blue}{\lambda_2 \cdot R - \lambda_1 \cdot R} \]

      4. distribute-rgt-out-- 33.2%

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

   8. Simplified 33.2%

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

   if 9.5000000000000009e-38 < phi2

   1. Initial program 50.6%

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

      1. hypot-def 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. Taylor expanded in phi1 around 0 92.6%

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

   5. Taylor expanded in phi2 around inf 61.5%

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

      1. +-commutative 61.5%

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

      2. mul-1-neg 61.5%

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

      3. unsub-neg 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-157}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 16: 25.6% accurate, 65.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.16e+35) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.16e+35) {
		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 <= 1.16d+35) 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 <= 1.16e+35) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.16e+35:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.16e+35)
		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 <= 1.16e+35)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.16e+35], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.1600000000000001e35

   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-def 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in phi1 around 0 88.0%

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

   5. Taylor expanded in phi2 around 0 86.0%

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

   6. Taylor expanded in lambda2 around inf 17.6%

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

   if 1.1600000000000001e35 < phi2

   1. Initial program 48.3%

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

      1. hypot-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in phi2 around inf 65.4%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.16 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 17: 13.6% 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 58.8%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in phi1 around 0 89.9%

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

5. Taylor expanded in phi2 around 0 86.6%

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

6. Taylor expanded in lambda2 around inf 14.8%

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

7. Final simplification 14.8%

   \[\leadsto R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2023189 
(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))))))