Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.9% → 99.8%

Time: 26.4s

Alternatives: 14

Speedup: 3.0×

• 30.5% 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: 59.9% 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: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

   1. add-sqr-sqrt 61.7%

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

   2. sqrt-unprod 97.1%

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

   3. sqr-cos-a 97.1%

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

   4. cos-2 97.1%

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

   5. cos-sum 97.1%

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

   6. add-log-exp 26.6%

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

   7. add-log-exp 26.6%

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

   8. sum-log 26.2%

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

   9. exp-sqrt 26.2%

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

   10. exp-sqrt 26.2%

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

   11. add-sqr-sqrt 26.2%

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

   12. add-log-exp 97.1%

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

5. Applied egg-rr 97.1%

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

   1. +-commutative 97.1%

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

7. Simplified 97.1%

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

   1. cos-sum 99.8%

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

9. Applied egg-rr 99.8%

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

    1. expm1-log1p-u 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \sqrt{0.5 + 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)}, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4.4e+152)
   (*
    R
    (hypot
     (*
      lambda1
      (sqrt
       (+
        0.5
        (* 0.5 (- (* (cos phi2) (cos phi1)) (* (sin phi2) (sin phi1)))))))
     (- phi1 phi2)))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.4e+152) {
		tmp = R * hypot((lambda1 * sqrt((0.5 + (0.5 * ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.4e+152) {
		tmp = R * Math.hypot((lambda1 * Math.sqrt((0.5 + (0.5 * ((Math.cos(phi2) * Math.cos(phi1)) - (Math.sin(phi2) * Math.sin(phi1))))))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -4.4e+152:
		tmp = R * math.hypot((lambda1 * math.sqrt((0.5 + (0.5 * ((math.cos(phi2) * math.cos(phi1)) - (math.sin(phi2) * math.sin(phi1))))))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -4.4e+152)
		tmp = Float64(R * hypot(Float64(lambda1 * sqrt(Float64(0.5 + Float64(0.5 * Float64(Float64(cos(phi2) * cos(phi1)) - Float64(sin(phi2) * sin(phi1))))))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -4.4e+152)
		tmp = R * hypot((lambda1 * sqrt((0.5 + (0.5 * ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -4.4e+152], N[(R * N[Sqrt[N[(lambda1 * N[Sqrt[N[(0.5 + N[(0.5 * N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if lambda1 < -4.3999999999999996e152

   1. Initial program 43.3%

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

      1. hypot-def 92.8%

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

   3. Simplified 92.8%

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

      1. add-sqr-sqrt 64.7%

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

      2. sqrt-unprod 92.8%

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

      3. sqr-cos-a 92.7%

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

      4. cos-2 92.7%

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

      5. cos-sum 92.7%

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

      6. add-log-exp 21.9%

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

      7. add-log-exp 21.9%

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

      8. sum-log 21.9%

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

      9. exp-sqrt 21.9%

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

      10. exp-sqrt 21.9%

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

      11. add-sqr-sqrt 21.9%

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

      12. add-log-exp 92.7%

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

   5. Applied egg-rr 92.7%

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

      1. +-commutative 92.7%

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

   7. Simplified 92.7%

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

      1. cos-sum 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in lambda1 around inf 88.3%

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

   if -4.3999999999999996e152 < lambda1

   1. Initial program 63.7%

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

      1. hypot-def 97.8%

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

   3. Simplified 97.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \sqrt{0.5 + 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)}, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

   1. add-sqr-sqrt 61.7%

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

   2. sqrt-unprod 97.1%

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

   3. sqr-cos-a 97.1%

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

   4. cos-2 97.1%

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

   5. cos-sum 97.1%

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

   6. add-log-exp 26.6%

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

   7. add-log-exp 26.6%

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

   8. sum-log 26.2%

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

   9. exp-sqrt 26.2%

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

   10. exp-sqrt 26.2%

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

   11. add-sqr-sqrt 26.2%

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

   12. add-log-exp 97.1%

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

5. Applied egg-rr 97.1%

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

   1. +-commutative 97.1%

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

7. Simplified 97.1%

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

   1. cos-sum 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 4: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.1:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.1)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.1) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.1) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.1:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.1)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.1)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.1], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.10000000000000001

   1. Initial program 51.4%

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

      1. hypot-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in phi2 around 0 92.6%

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

   if -0.10000000000000001 < phi1

   1. Initial program 64.2%

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

      1. hypot-def 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in phi1 around 0 94.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.1:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5: 95.6% 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 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

4. Final simplification 97.1%

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

Alternative 6: 84.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.22 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \cos \phi_1} \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.22e+212)
   (* (sqrt (+ 0.5 (* 0.5 (cos phi1)))) (* R (- lambda1)))
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.22e+212) {
		tmp = sqrt((0.5 + (0.5 * cos(phi1)))) * (R * -lambda1);
	} 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 (lambda1 <= -1.22e+212) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.cos(phi1)))) * (R * -lambda1);
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.22e+212:
		tmp = math.sqrt((0.5 + (0.5 * math.cos(phi1)))) * (R * -lambda1)
	else:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.22e+212)
		tmp = Float64(sqrt(Float64(0.5 + Float64(0.5 * cos(phi1)))) * Float64(R * Float64(-lambda1)));
	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 (lambda1 <= -1.22e+212)
		tmp = sqrt((0.5 + (0.5 * cos(phi1)))) * (R * -lambda1);
	else
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.22e+212], N[(N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(R * (-lambda1)), $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 lambda1 < -1.22000000000000005e212

   1. Initial program 49.8%

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

      1. hypot-def 88.4%

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

   3. Simplified 88.4%

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

      1. add-sqr-sqrt 62.2%

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

      2. sqrt-unprod 88.4%

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

      3. sqr-cos-a 88.3%

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

      4. cos-2 88.3%

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

      5. cos-sum 88.3%

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

      6. add-log-exp 21.1%

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

      7. add-log-exp 21.1%

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

      8. sum-log 21.1%

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

      9. exp-sqrt 21.1%

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

      10. exp-sqrt 21.1%

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

      11. add-sqr-sqrt 21.1%

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

      12. add-log-exp 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in phi2 around 0 84.0%

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

      1. *-commutative 84.0%

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

   10. Simplified 84.0%

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

   11. Taylor expanded in lambda1 around -inf 83.5%

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

       1. associate-*r* 83.5%

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

       2. mul-1-neg 83.5%

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

       3. *-commutative 83.5%

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

   13. Simplified 83.5%

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

   if -1.22000000000000005e212 < lambda1

   1. Initial program 62.0%

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

      1. hypot-def 97.8%

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

   3. Simplified 97.8%

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

      1. add-sqr-sqrt 61.7%

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

      2. sqrt-unprod 97.8%

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

      3. sqr-cos-a 97.8%

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

      4. cos-2 97.8%

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

      5. cos-sum 97.8%

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

      6. add-log-exp 27.0%

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

      7. add-log-exp 27.0%

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

      8. sum-log 26.6%

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

      9. exp-sqrt 26.6%

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

      10. exp-sqrt 26.6%

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

      11. add-sqr-sqrt 26.6%

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

      12. add-log-exp 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. +-commutative 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in phi2 around 0 92.5%

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

      1. *-commutative 92.5%

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

   10. Simplified 92.5%

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

   11. Taylor expanded in phi1 around 0 86.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.22 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \cos \phi_1} \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in phi2 around 0 91.9%

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

5. Final simplification 91.9%

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

Alternative 8: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.55 \cdot 10^{+221}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \cos \phi_1} \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 2.55e+221)
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))
   (* (sqrt (+ 0.5 (* 0.5 (cos phi1)))) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.55e+221) {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = sqrt((0.5 + (0.5 * cos(phi1)))) * (R * lambda2);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.55e+221) {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * Math.cos(phi1)))) * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.55e+221:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	else:
		tmp = math.sqrt((0.5 + (0.5 * math.cos(phi1)))) * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.55e+221)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	else
		tmp = Float64(sqrt(Float64(0.5 + Float64(0.5 * cos(phi1)))) * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.55e+221)
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	else
		tmp = sqrt((0.5 + (0.5 * cos(phi1)))) * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 2.55e+221], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.54999999999999988e221

   1. Initial program 63.1%

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

      1. hypot-def 97.6%

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

   3. Simplified 97.6%

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

      1. add-sqr-sqrt 62.7%

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

      2. sqrt-unprod 97.6%

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

      3. sqr-cos-a 97.6%

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

      4. cos-2 97.6%

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

      5. cos-sum 97.6%

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

      6. add-log-exp 26.4%

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

      7. add-log-exp 26.4%

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

      8. sum-log 25.9%

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

      9. exp-sqrt 25.9%

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

      10. exp-sqrt 25.9%

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

      11. add-sqr-sqrt 25.9%

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

      12. add-log-exp 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in phi2 around 0 92.6%

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

      1. *-commutative 92.6%

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

   10. Simplified 92.6%

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

   11. Taylor expanded in phi1 around 0 86.9%

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

   if 2.54999999999999988e221 < lambda2

   1. Initial program 33.1%

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

      1. hypot-def 90.4%

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

   3. Simplified 90.4%

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

      1. add-sqr-sqrt 48.2%

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

      2. sqrt-unprod 90.4%

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

      3. sqr-cos-a 90.4%

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

      4. cos-2 90.3%

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

      5. cos-sum 90.4%

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

      6. add-log-exp 29.4%

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

      7. add-log-exp 29.4%

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

      8. sum-log 29.4%

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

      9. exp-sqrt 29.4%

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

      10. exp-sqrt 29.4%

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

      11. add-sqr-sqrt 29.4%

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

      12. add-log-exp 90.4%

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

   5. Applied egg-rr 90.4%

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

      1. +-commutative 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in phi2 around 0 81.8%

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

      1. *-commutative 81.8%

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

   10. Simplified 81.8%

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

   11. Taylor expanded in lambda2 around inf 81.9%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.55 \cdot 10^{+221}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \cos \phi_1} \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]

Alternative 9: 70.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{+138}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.45e+138)
   (* R (- phi2 phi1))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.45e+138) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.45e+138) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.45e+138:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.45e+138)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.45e+138)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.45e+138], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.45000000000000005e138

   1. Initial program 49.1%

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

      1. hypot-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in phi1 around -inf 76.6%

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

      1. mul-1-neg 76.6%

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

      2. unsub-neg 76.6%

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

   6. Simplified 76.6%

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

   if -1.45000000000000005e138 < phi1

   1. Initial program 62.6%

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

      1. hypot-def 97.4%

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

   3. Simplified 97.4%

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

      1. add-sqr-sqrt 63.5%

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

      2. sqrt-unprod 97.4%

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

      3. sqr-cos-a 97.4%

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

      4. cos-2 97.4%

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

      5. cos-sum 97.4%

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

      6. add-log-exp 29.8%

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

      7. add-log-exp 29.8%

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

      8. sum-log 29.4%

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

      9. exp-sqrt 29.4%

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

      10. exp-sqrt 29.4%

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

      11. add-sqr-sqrt 29.4%

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

      12. add-log-exp 97.4%

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

   5. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in phi2 around 0 91.5%

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

      1. *-commutative 91.5%

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

   10. Simplified 91.5%

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

   11. Taylor expanded in phi1 around 0 49.3%

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

       1. unpow2 49.3%

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

       2. unpow2 49.3%

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

       3. hypot-def 69.9%

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

   13. Simplified 69.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{+138}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]

Alternative 10: 84.4% 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 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

   1. add-sqr-sqrt 61.7%

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

   2. sqrt-unprod 97.1%

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

   3. sqr-cos-a 97.1%

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

   4. cos-2 97.1%

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

   5. cos-sum 97.1%

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

   6. add-log-exp 26.6%

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

   7. add-log-exp 26.6%

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

   8. sum-log 26.2%

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

   9. exp-sqrt 26.2%

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

   10. exp-sqrt 26.2%

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

   11. add-sqr-sqrt 26.2%

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

   12. add-log-exp 97.1%

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

5. Applied egg-rr 97.1%

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

   1. +-commutative 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in phi2 around 0 91.9%

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

   1. *-commutative 91.9%

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

10. Simplified 91.9%

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

11. Taylor expanded in phi1 around 0 85.4%

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

12. Final simplification 85.4%

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

Alternative 11: 26.9% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{+111}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-136} \lor \neg \left(\phi_1 \leq -1.75 \cdot 10^{-202}\right) \land \phi_1 \leq 8.2 \cdot 10^{-282}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.05e+111)
   (* R (- phi1))
   (if (or (<= phi1 -1e-136)
           (and (not (<= phi1 -1.75e-202)) (<= phi1 8.2e-282)))
     (* R (- lambda1))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.05e+111) {
		tmp = R * -phi1;
	} else if ((phi1 <= -1e-136) || (!(phi1 <= -1.75e-202) && (phi1 <= 8.2e-282))) {
		tmp = R * -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 (phi1 <= (-2.05d+111)) then
        tmp = r * -phi1
    else if ((phi1 <= (-1d-136)) .or. (.not. (phi1 <= (-1.75d-202))) .and. (phi1 <= 8.2d-282)) then
        tmp = r * -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 (phi1 <= -2.05e+111) {
		tmp = R * -phi1;
	} else if ((phi1 <= -1e-136) || (!(phi1 <= -1.75e-202) && (phi1 <= 8.2e-282))) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.05e+111:
		tmp = R * -phi1
	elif (phi1 <= -1e-136) or (not (phi1 <= -1.75e-202) and (phi1 <= 8.2e-282)):
		tmp = R * -lambda1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.05e+111)
		tmp = Float64(R * Float64(-phi1));
	elseif ((phi1 <= -1e-136) || (!(phi1 <= -1.75e-202) && (phi1 <= 8.2e-282)))
		tmp = Float64(R * Float64(-lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.05e+111)
		tmp = R * -phi1;
	elseif ((phi1 <= -1e-136) || (~((phi1 <= -1.75e-202)) && (phi1 <= 8.2e-282)))
		tmp = R * -lambda1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.05e+111], N[(R * (-phi1)), $MachinePrecision], If[Or[LessEqual[phi1, -1e-136], And[N[Not[LessEqual[phi1, -1.75e-202]], $MachinePrecision], LessEqual[phi1, 8.2e-282]]], N[(R * (-lambda1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.04999999999999993e111

   1. Initial program 47.5%

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

      1. hypot-def 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in phi1 around -inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

   6. Simplified 76.5%

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

   if -2.04999999999999993e111 < phi1 < -1e-136 or -1.75e-202 < phi1 < 8.19999999999999954e-282

   1. Initial program 65.3%

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

      1. hypot-def 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in lambda1 around -inf 14.3%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 14.3%

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

      2. *-commutative 14.3%

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

      3. distribute-rgt-neg-in 14.3%

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

   6. Simplified 14.3%

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

   7. Taylor expanded in phi2 around 0 18.6%

      \[\leadsto \left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)} \cdot \lambda_1\right) \cdot \left(-R\right) \]
   8. Step-by-step derivation

      1. *-commutative 18.6%

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

      2. *-commutative 18.6%

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

   9. Simplified 18.6%

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

   10. Taylor expanded in phi1 around 0 13.8%

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

   if -1e-136 < phi1 < -1.75e-202 or 8.19999999999999954e-282 < phi1

   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 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in phi2 around inf 20.1%

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

      1. *-commutative 20.1%

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

   6. Simplified 20.1%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{+111}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-136} \lor \neg \left(\phi_1 \leq -1.75 \cdot 10^{-202}\right) \land \phi_1 \leq 8.2 \cdot 10^{-282}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 34.9% accurate, 46.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.3 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.3e+152) (* R (- lambda1)) (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.3e+152) {
		tmp = R * -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 (lambda1 <= (-1.3d+152)) then
        tmp = r * -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 (lambda1 <= -1.3e+152) {
		tmp = R * -lambda1;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.3e+152:
		tmp = R * -lambda1
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.3e+152)
		tmp = Float64(R * Float64(-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 (lambda1 <= -1.3e+152)
		tmp = R * -lambda1;
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.3e+152], N[(R * (-lambda1)), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.3e152

   1. Initial program 43.3%

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

      1. hypot-def 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in lambda1 around -inf 53.1%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 53.1%

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

      2. *-commutative 53.1%

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

      3. distribute-rgt-neg-in 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in phi2 around 0 49.4%

      \[\leadsto \left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)} \cdot \lambda_1\right) \cdot \left(-R\right) \]
   8. Step-by-step derivation

      1. *-commutative 49.4%

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

      2. *-commutative 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in phi1 around 0 58.0%

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

   if -1.3e152 < lambda1

   1. Initial program 63.7%

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

      1. hypot-def 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in phi1 around -inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. unsub-neg 28.7%

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

   6. Simplified 28.7%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.3 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 13: 26.7% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.2e-9) (* R (- lambda1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e-9) {
		tmp = R * -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 <= 2.2d-9) then
        tmp = r * -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 <= 2.2e-9) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.2e-9:
		tmp = R * -lambda1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.2e-9)
		tmp = Float64(R * Float64(-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 <= 2.2e-9)
		tmp = R * -lambda1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.2e-9], N[(R * (-lambda1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1999999999999998e-9

   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 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in lambda1 around -inf 17.3%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 17.3%

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

      2. *-commutative 17.3%

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

      3. distribute-rgt-neg-in 17.3%

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

   6. Simplified 17.3%

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

   7. Taylor expanded in phi2 around 0 18.0%

      \[\leadsto \left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)} \cdot \lambda_1\right) \cdot \left(-R\right) \]
   8. Step-by-step derivation

      1. *-commutative 18.0%

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

      2. *-commutative 18.0%

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

   9. Simplified 18.0%

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

   10. Taylor expanded in phi1 around 0 15.4%

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

   if 2.1999999999999998e-9 < phi2

   1. Initial program 56.4%

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

      1. hypot-def 95.1%

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

   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 phi2 around inf 63.3%

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

      1. *-commutative 63.3%

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

   6. Simplified 63.3%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14: 17.9% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. hypot-def 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in phi2 around inf 18.9%

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

   1. *-commutative 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023279 
(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))))))