Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.8% → 99.4%

Time: 18.3s

Alternatives: 15

Speedup: 3.0×

• 30.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot {\left(\sqrt{R\_m \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (pow
   (sqrt
    (*
     R_m
     (hypot
      (*
       (-
        (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
        (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
       (- lambda1 lambda2))
      (- phi1 phi2))))
   2.0)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * pow(sqrt((R_m * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))), 2.0);
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * Math.pow(Math.sqrt((R_m * Math.hypot((((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))), 2.0);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * math.pow(math.sqrt((R_m * math.hypot((((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))), 2.0)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * (sqrt(Float64(R_m * hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))) ^ 2.0))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (sqrt((R_m * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))) ^ 2.0);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. hypot-define 96.4%

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

3. Simplified 96.4%

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

   1. add-sqr-sqrt 51.1%

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

   2. pow2 51.1%

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

   3. *-commutative 51.1%

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

   4. div-inv 51.1%

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

   5. metadata-eval 51.1%

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

6. Applied egg-rr 51.1%

   \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. distribute-rgt-in 51.1%

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

   4. *-commutative 51.1%

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

   5. cos-sum 52.4%

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

   6. *-commutative 52.4%

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

   7. *-commutative 52.4%

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

8. Applied egg-rr 52.4%

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

9. Final simplification 52.4%

   \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.5 \cdot 10^{+212}:\\ \;\;\;\;{\left(\sqrt{R\_m \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1, \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -3.5e+212)
    (pow
     (sqrt
      (*
       R_m
       (hypot
        (*
         (-
          (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
          (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
         lambda1)
        (- phi1 phi2))))
     2.0)
    (*
     R_m
     (hypot
      (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
      (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.5e+212) {
		tmp = pow(sqrt((R_m * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * lambda1), (phi1 - phi2)))), 2.0);
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.5e+212) {
		tmp = Math.pow(Math.sqrt((R_m * Math.hypot((((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) * lambda1), (phi1 - phi2)))), 2.0);
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3.5e+212:
		tmp = math.pow(math.sqrt((R_m * math.hypot((((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) * lambda1), (phi1 - phi2)))), 2.0)
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.5e+212)
		tmp = sqrt(Float64(R_m * hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) * lambda1), Float64(phi1 - phi2)))) ^ 2.0;
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3.5e+212)
		tmp = sqrt((R_m * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * lambda1), (phi1 - phi2)))) ^ 2.0;
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -3.5e+212], N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], N[(R$95$m * 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.49999999999999987e212

   1. Initial program 32.5%

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

      1. hypot-define 80.4%

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

   3. Simplified 80.4%

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

      1. add-sqr-sqrt 49.2%

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

      2. pow2 49.2%

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

      3. *-commutative 49.2%

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

      4. div-inv 49.2%

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

      5. metadata-eval 49.2%

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

   6. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 49.2%

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

      2. +-commutative 49.2%

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

      3. distribute-rgt-in 49.2%

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

      4. *-commutative 49.2%

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

      5. cos-sum 56.8%

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

      6. *-commutative 56.8%

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

      7. *-commutative 56.8%

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

   8. Applied egg-rr 56.8%

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

   9. Taylor expanded in lambda1 around inf 56.8%

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

   if -3.49999999999999987e212 < lambda1

   1. Initial program 62.8%

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

      1. hypot-define 97.8%

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

   3. Simplified 97.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.5 \cdot 10^{+212}:\\ \;\;\;\;{\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1, \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -3.5e-7)
    (* R_m (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-7) {
		tmp = R_m * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-7) {
		tmp = R_m * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.5e-7:
		tmp = R_m * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.5e-7)
		tmp = Float64(R_m * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.5e-7)
		tmp = R_m * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -3.5e-7], N[(R$95$m * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.49999999999999984e-7

   1. Initial program 57.8%

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

      1. hypot-define 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in phi2 around 0 93.1%

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

   if -3.49999999999999984e-7 < 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-define 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in phi1 around 0 93.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-211}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 1e-211)
    (* R_m (hypot (* lambda1 (cos (* phi2 0.5))) (- phi1 phi2)))
    (* R_m (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-211) {
		tmp = R_m * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-211) {
		tmp = R_m * Math.hypot((lambda1 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1e-211:
		tmp = R_m * math.hypot((lambda1 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1e-211)
		tmp = Float64(R_m * hypot(Float64(lambda1 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1e-211)
		tmp = R_m * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 1e-211], N[(R$95$m * N[Sqrt[N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.00000000000000009e-211

   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-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi1 around 0 93.9%

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

   6. Taylor expanded in lambda1 around inf 82.0%

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

   if 1.00000000000000009e-211 < lambda2

   1. Initial program 59.3%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around 0 89.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-211}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 7.2e+49)
    (* R_m (hypot (* lambda1 (cos (* phi2 0.5))) (- phi1 phi2)))
    (* R_m (hypot phi2 (- lambda1 lambda2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 7.2e+49) {
		tmp = R_m * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 7.2e+49) {
		tmp = R_m * Math.hypot((lambda1 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 7.2e+49:
		tmp = R_m * math.hypot((lambda1 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot(phi2, (lambda1 - lambda2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 7.2e+49)
		tmp = Float64(R_m * hypot(Float64(lambda1 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 7.2e+49)
		tmp = R_m * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 7.2e+49], N[(R$95$m * N[Sqrt[N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 7.19999999999999993e49

   1. Initial program 61.5%

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

      1. hypot-define 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 94.9%

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

   6. Taylor expanded in lambda1 around inf 85.7%

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

   if 7.19999999999999993e49 < lambda2

   1. Initial program 56.7%

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

      1. hypot-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in phi2 around 0 74.4%

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

      1. associate-*r* 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in phi1 around 0 49.0%

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

      1. unpow2 49.0%

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

      2. unpow2 49.0%

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

      3. hypot-define 62.8%

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

   10. Simplified 62.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   R_m
   (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * 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]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. hypot-define 96.4%

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

3. Simplified 96.4%

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

5. Final simplification 96.4%

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

Alternative 7: 70.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -2.5e+14)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (* R_m (hypot phi2 (- lambda1 lambda2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e+14) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e+14) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.5e+14:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	else:
		tmp = R_m * math.hypot(phi2, (lambda1 - lambda2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.5e+14)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	else
		tmp = Float64(R_m * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.5e+14)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	else
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -2.5e+14], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.5e14

   1. Initial program 59.6%

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

      1. hypot-define 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in phi1 around -inf 67.6%

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

      1. associate-*r* 67.6%

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

      2. mul-1-neg 67.6%

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

      3. mul-1-neg 67.6%

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

      4. unsub-neg 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-/l* 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in phi2 around 0 67.6%

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

      1. associate-/l* 72.1%

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

   10. Simplified 72.1%

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

   if -2.5e14 < phi1

   1. Initial program 60.6%

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

      1. hypot-define 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in phi2 around 0 81.3%

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

      1. associate-*r* 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in phi1 around 0 47.5%

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

      1. unpow2 47.5%

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

      2. unpow2 47.5%

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

      3. hypot-define 67.9%

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

   10. Simplified 67.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{+242}:\\ \;\;\;\;R\_m \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.45 \cdot 10^{-306}:\\ \;\;\;\;R\_m \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \phi_1 \cdot \frac{R\_m}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.7e+242)
    (* R_m (- lambda1))
    (if (<= lambda1 -1.45e-306)
      (* R_m (* phi1 (+ (/ phi2 phi1) -1.0)))
      (* phi2 (- R_m (* phi1 (/ R_m phi2))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.7e+242) {
		tmp = R_m * -lambda1;
	} else if (lambda1 <= -1.45e-306) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-1.7d+242)) then
        tmp = r_m * -lambda1
    else if (lambda1 <= (-1.45d-306)) then
        tmp = r_m * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else
        tmp = phi2 * (r_m - (phi1 * (r_m / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.7e+242) {
		tmp = R_m * -lambda1;
	} else if (lambda1 <= -1.45e-306) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.7e+242:
		tmp = R_m * -lambda1
	elif lambda1 <= -1.45e-306:
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.7e+242)
		tmp = Float64(R_m * Float64(-lambda1));
	elseif (lambda1 <= -1.45e-306)
		tmp = Float64(R_m * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(phi2 * Float64(R_m - Float64(phi1 * Float64(R_m / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.7e+242)
		tmp = R_m * -lambda1;
	elseif (lambda1 <= -1.45e-306)
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -1.7e+242], N[(R$95$m * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, -1.45e-306], N[(R$95$m * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R$95$m - N[(phi1 * N[(R$95$m / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.69999999999999991e242

   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-define 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in lambda1 around -inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. associate-*r* 44.4%

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

      3. distribute-lft-neg-in 44.4%

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

      4. +-commutative 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in phi2 around 0 47.9%

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

   9. Taylor expanded in phi1 around 0 56.6%

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

       1. mul-1-neg 56.6%

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

   11. Simplified 56.6%

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

   if -1.69999999999999991e242 < lambda1 < -1.4499999999999999e-306

   1. Initial program 64.8%

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

      1. hypot-define 97.5%

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

   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. Add Preprocessing

   5. Taylor expanded in phi1 around -inf 29.4%

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

      1. mul-1-neg 29.4%

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

      2. distribute-rgt-neg-in 29.4%

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

      3. mul-1-neg 29.4%

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

      4. unsub-neg 29.4%

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

   7. Simplified 29.4%

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

   if -1.4499999999999999e-306 < lambda1

   1. Initial program 60.3%

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

      1. hypot-define 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. Add Preprocessing

   5. Taylor expanded in phi2 around inf 35.5%

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

      1. mul-1-neg 35.5%

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

      2. unsub-neg 35.5%

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

      3. *-commutative 35.5%

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

      4. associate-/l* 32.2%

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

   7. Simplified 32.2%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{+242}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.45 \cdot 10^{-306}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-246}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \phi_1 \cdot \frac{R\_m}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -1.9e-246)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (* phi2 (- R_m (* phi1 (/ R_m phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.9e-246) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-1.9d-246)) then
        tmp = phi1 * ((r_m * (phi2 / phi1)) - r_m)
    else
        tmp = phi2 * (r_m - (phi1 * (r_m / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.9e-246) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.9e-246:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	else:
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.9e-246)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	else
		tmp = Float64(phi2 * Float64(R_m - Float64(phi1 * Float64(R_m / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.9e-246)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	else
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -1.9e-246], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R$95$m - N[(phi1 * N[(R$95$m / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.89999999999999988e-246

   1. Initial program 57.3%

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

      1. hypot-define 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in phi1 around -inf 52.2%

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

      1. associate-*r* 52.2%

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

      2. mul-1-neg 52.2%

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

      3. mul-1-neg 52.2%

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

      4. unsub-neg 52.2%

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

      5. *-commutative 52.2%

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

      6. associate-/l* 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in phi2 around 0 52.2%

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

      1. associate-/l* 54.8%

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

   10. Simplified 54.8%

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

   if -1.89999999999999988e-246 < phi1

   1. Initial program 62.8%

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

      1. hypot-define 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi2 around inf 16.2%

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

      1. mul-1-neg 16.2%

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

      2. unsub-neg 16.2%

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

      3. *-commutative 16.2%

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

      4. associate-/l* 16.5%

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

   7. Simplified 16.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-246}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.1% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.15 \cdot 10^{+223}:\\ \;\;\;\;R\_m \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \phi_1 \cdot \frac{R\_m}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.15e+223)
    (* R_m (- lambda1))
    (* phi2 (- R_m (* phi1 (/ R_m phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.15e+223) {
		tmp = R_m * -lambda1;
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-1.15d+223)) then
        tmp = r_m * -lambda1
    else
        tmp = phi2 * (r_m - (phi1 * (r_m / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.15e+223) {
		tmp = R_m * -lambda1;
	} else {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.15e+223:
		tmp = R_m * -lambda1
	else:
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.15e+223)
		tmp = Float64(R_m * Float64(-lambda1));
	else
		tmp = Float64(phi2 * Float64(R_m - Float64(phi1 * Float64(R_m / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.15e+223)
		tmp = R_m * -lambda1;
	else
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -1.15e+223], N[(R$95$m * (-lambda1)), $MachinePrecision], N[(phi2 * N[(R$95$m - N[(phi1 * N[(R$95$m / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.15000000000000002e223

   1. Initial program 35.1%

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

      1. hypot-define 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in lambda1 around -inf 40.1%

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

      1. mul-1-neg 40.1%

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

      2. associate-*r* 40.1%

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

      3. distribute-lft-neg-in 40.1%

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

      4. +-commutative 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in phi2 around 0 48.2%

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

   9. Taylor expanded in phi1 around 0 56.3%

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

       1. mul-1-neg 56.3%

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

   11. Simplified 56.3%

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

   if -1.15000000000000002e223 < lambda1

   1. Initial program 62.4%

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

      1. hypot-define 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. Add Preprocessing

   5. Taylor expanded in phi2 around inf 32.6%

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

      1. mul-1-neg 32.6%

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

      2. unsub-neg 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-/l* 30.6%

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

   7. Simplified 30.6%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.15 \cdot 10^{+223}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-54}:\\ \;\;\;\;R\_m \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 2.9e-54)
    (* R_m (- phi1))
    (* R_m (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.9e-54) {
		tmp = R_m * -phi1;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= 2.9d-54) then
        tmp = r_m * -phi1
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.9e-54) {
		tmp = R_m * -phi1;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.9e-54:
		tmp = R_m * -phi1
	else:
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.9e-54)
		tmp = Float64(R_m * Float64(-phi1));
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.9e-54)
		tmp = R_m * -phi1;
	else
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 2.9e-54], N[(R$95$m * (-phi1)), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.90000000000000015e-54

   1. Initial program 63.3%

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

      1. hypot-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in phi1 around -inf 24.1%

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

      1. mul-1-neg 24.1%

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

   7. Simplified 24.1%

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

   if 2.90000000000000015e-54 < phi2

   1. Initial program 53.0%

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

      1. hypot-define 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in phi2 around inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. unsub-neg 58.1%

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

   7. Simplified 58.1%

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

Alternative 12: 28.5% accurate, 36.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -17000000000:\\ \;\;\;\;R\_m \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi1 -17000000000.0) (* R_m (- phi1)) (* R_m phi2))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -17000000000.0) {
		tmp = R_m * -phi1;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-17000000000.0d0)) then
        tmp = r_m * -phi1
    else
        tmp = r_m * phi2
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -17000000000.0) {
		tmp = R_m * -phi1;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -17000000000.0:
		tmp = R_m * -phi1
	else:
		tmp = R_m * phi2
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -17000000000.0)
		tmp = Float64(R_m * Float64(-phi1));
	else
		tmp = Float64(R_m * phi2);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -17000000000.0)
		tmp = R_m * -phi1;
	else
		tmp = R_m * phi2;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -17000000000.0], N[(R$95$m * (-phi1)), $MachinePrecision], N[(R$95$m * phi2), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7e10

   1. Initial program 59.6%

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

      1. hypot-define 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in phi1 around -inf 73.7%

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

      1. mul-1-neg 73.7%

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

   7. Simplified 73.7%

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

   if -1.7e10 < phi1

   1. Initial program 60.6%

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

      1. hypot-define 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in phi2 around inf 18.3%

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

      1. *-commutative 18.3%

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

   7. Simplified 18.3%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -17000000000:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 25.9% accurate, 36.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{-36}:\\ \;\;\;\;R\_m \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi2 3.2e-36) (* R_m (- lambda1)) (* R_m phi2))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.2e-36) {
		tmp = R_m * -lambda1;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= 3.2d-36) then
        tmp = r_m * -lambda1
    else
        tmp = r_m * phi2
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.2e-36) {
		tmp = R_m * -lambda1;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.2e-36:
		tmp = R_m * -lambda1
	else:
		tmp = R_m * phi2
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.2e-36)
		tmp = Float64(R_m * Float64(-lambda1));
	else
		tmp = Float64(R_m * phi2);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.2e-36)
		tmp = R_m * -lambda1;
	else
		tmp = R_m * phi2;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 3.2e-36], N[(R$95$m * (-lambda1)), $MachinePrecision], N[(R$95$m * phi2), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.20000000000000021e-36

   1. Initial program 64.0%

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

      1. hypot-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in lambda1 around -inf 15.4%

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

      1. mul-1-neg 15.4%

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

      2. associate-*r* 15.4%

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

      3. distribute-lft-neg-in 15.4%

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

      4. +-commutative 15.4%

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

   7. Simplified 15.4%

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

   8. Taylor expanded in phi2 around 0 15.4%

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

   9. Taylor expanded in phi1 around 0 13.5%

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

       1. mul-1-neg 13.5%

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

   11. Simplified 13.5%

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

   if 3.20000000000000021e-36 < phi2

   1. Initial program 50.3%

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

      1. hypot-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in phi2 around inf 47.4%

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

      1. *-commutative 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 17.4% accurate, 109.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \phi_2\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m phi2)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * phi2);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (r_m * phi2)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * phi2);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * phi2)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * phi2))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * phi2);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * phi2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. hypot-define 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in phi2 around inf 15.8%

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

   1. *-commutative 15.8%

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

7. Simplified 15.8%

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

8. Final simplification 15.8%

   \[\leadsto R \cdot \phi_2 \]
9. Add Preprocessing

Alternative 15: 14.1% accurate, 109.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \lambda_1\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m lambda1)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * lambda1);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (r_m * lambda1)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * lambda1);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * lambda1)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * lambda1))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * lambda1);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * lambda1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. hypot-define 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in lambda1 around -inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. associate-*r* 13.9%

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

   3. distribute-lft-neg-in 13.9%

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

   4. +-commutative 13.9%

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

7. Simplified 13.9%

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

8. Taylor expanded in phi2 around 0 13.7%

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

9. Taylor expanded in phi1 around 0 12.0%

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

    1. mul-1-neg 12.0%

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

11. Simplified 12.0%

    \[\leadsto \color{blue}{-R \cdot \lambda_1} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 7.6%

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

    2. sqrt-unprod 14.1%

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

    3. sqr-neg 14.1%

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

    4. sqrt-unprod 8.8%

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

    5. add-sqr-sqrt 14.5%

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

    6. *-commutative 14.5%

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

13. Applied egg-rr 14.5%

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

14. Final simplification 14.5%

    \[\leadsto R \cdot \lambda_1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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))))))