Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.5% → 99.4%

Time: 23.3s

Alternatives: 10

Speedup: 3.0×

• 31.4% 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: 60.5% 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

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

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (pow
   (sqrt
    (*
     R_m
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
        (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
      (- 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(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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(((lambda1 - lambda2) * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (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(((lambda1 - lambda2) * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (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(lambda1 - lambda2) * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)))) ^ 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(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

   1. hypot-def 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. Step-by-step derivation

   1. add-sqr-sqrt 49.6%

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

   2. pow2 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. metadata-eval 49.6%

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

6. Applied egg-rr 49.6%

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

   1. *-commutative 49.6%

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

   2. +-commutative 49.6%

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

   3. distribute-lft-in 49.6%

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

   4. cos-sum 52.5%

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

   5. *-commutative 52.5%

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

   6. *-commutative 52.5%

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

   7. *-commutative 52.5%

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

   8. *-commutative 52.5%

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

8. Applied egg-rr 52.5%

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

9. Final simplification 52.5%

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

Alternative 2: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{+212}:\\ \;\;\;\;{\left(\sqrt{R_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.05e+212)
    (pow
     (sqrt
      (*
       R_m
       (hypot
        (*
         lambda1
         (-
          (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
          (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
        (- 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 <= -1.05e+212) {
		tmp = pow(sqrt((R_m * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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 <= -1.05e+212) {
		tmp = Math.pow(Math.sqrt((R_m * Math.hypot((lambda1 * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (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 <= -1.05e+212:
		tmp = math.pow(math.sqrt((R_m * math.hypot((lambda1 * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (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 <= -1.05e+212)
		tmp = sqrt(Float64(R_m * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)))) ^ 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 <= -1.05e+212)
		tmp = sqrt((R_m * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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, -1.05e+212], N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $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 < -1.05e212

   1. Initial program 40.0%

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

      1. hypot-def 81.2%

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

   3. Simplified 81.2%

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

      1. add-sqr-sqrt 34.8%

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

      2. pow2 34.8%

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

      3. *-commutative 34.8%

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

      4. div-inv 34.8%

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

      5. metadata-eval 34.8%

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

   6. Applied egg-rr 34.8%

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

      1. *-commutative 34.8%

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

      2. +-commutative 34.8%

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

      3. distribute-lft-in 34.8%

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

      4. cos-sum 49.5%

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

      5. *-commutative 49.5%

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

      6. *-commutative 49.5%

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

      7. *-commutative 49.5%

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

      8. *-commutative 49.5%

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

   8. Applied egg-rr 49.5%

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

   9. Taylor expanded in lambda1 around inf 44.0%

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

   if -1.05e212 < lambda1

   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-def 97.2%

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

   3. Simplified 97.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{+212}:\\ \;\;\;\;{\left(\sqrt{R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-137}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 2.6e-137)
    (* R_m (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
    (* 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 (phi2 <= 2.6e-137) {
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} 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 (phi2 <= 2.6e-137) {
		tmp = R_m * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} 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 phi2 <= 2.6e-137:
		tmp = R_m * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	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 (phi2 <= 2.6e-137)
		tmp = Float64(R_m * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	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 (phi2 <= 2.6e-137)
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	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[phi2, 2.6e-137], N[(R$95$m * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $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 phi2 < 2.6e-137

   1. Initial program 66.2%

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

      1. hypot-def 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in phi2 around 0 59.2%

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

      1. +-commutative 59.2%

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

      2. unpow2 59.2%

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

      3. unpow2 59.2%

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

      4. unpow2 59.2%

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

      5. unswap-sqr 59.2%

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

      6. hypot-def 81.6%

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

   7. Simplified 81.6%

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

   if 2.6e-137 < phi2

   1. Initial program 50.0%

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

      1. hypot-def 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 phi1 around 0 89.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-137}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 2.05e+63)
    (* R_m (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
    (* R_m (- phi2 phi1)))))

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 <= 2.05e+63) {
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R_m * (phi2 - phi1);
	}
	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 (phi2 <= 2.05e+63) {
		tmp = R_m * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R_m * (phi2 - phi1);
	}
	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 <= 2.05e+63:
		tmp = R_m * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R_m * (phi2 - phi1)
	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 <= 2.05e+63)
		tmp = Float64(R_m * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R_m * Float64(phi2 - phi1));
	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 <= 2.05e+63)
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R_m * (phi2 - phi1);
	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, 2.05e+63], N[(R$95$m * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.04999999999999996e63

   1. Initial program 65.9%

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

      1. hypot-def 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. unpow2 58.2%

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

      3. unpow2 58.2%

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

      4. unpow2 58.2%

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

      5. unswap-sqr 58.2%

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

      6. hypot-def 81.0%

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

   7. Simplified 81.0%

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

   if 2.04999999999999996e63 < phi2

   1. Initial program 36.1%

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

      1. hypot-def 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in phi1 around -inf 70.8%

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

      1. +-commutative 70.8%

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

      2. mul-1-neg 70.8%

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

      3. unsub-neg 70.8%

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

      4. *-commutative 70.8%

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

      5. *-commutative 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in R around -inf 70.8%

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

      1. associate-*r* 70.8%

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

      2. neg-mul-1 70.8%

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

      3. distribute-lft-out-- 70.8%

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

      4. neg-mul-1 70.8%

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

      5. distribute-lft-neg-in 70.8%

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

      6. neg-mul-1 70.8%

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

      7. distribute-lft-out-- 70.8%

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

      8. cancel-sign-sub-inv 70.8%

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

      9. metadata-eval 70.8%

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

      10. *-lft-identity 70.8%

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

      11. distribute-lft-in 70.8%

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

      12. mul-1-neg 70.8%

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

      13. distribute-rgt-neg-in 70.8%

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

      14. mul-1-neg 70.8%

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

      15. +-commutative 70.8%

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

      16. mul-1-neg 70.8%

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

      17. distribute-rgt-neg-in 70.8%

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

      18. mul-1-neg 70.8%

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

      19. distribute-lft-in 70.8%

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

      20. mul-1-neg 70.8%

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

      21. sub-neg 70.8%

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

      22. *-commutative 70.8%

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

   10. Simplified 70.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -3800.0)
    (* R_m (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
    (* R_m (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5))))))))

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 <= -3800.0) {
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R_m * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	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 <= -3800.0) {
		tmp = R_m * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R_m * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	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 <= -3800.0:
		tmp = R_m * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R_m * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	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 <= -3800.0)
		tmp = Float64(R_m * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R_m * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	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 <= -3800.0)
		tmp = R_m * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R_m * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	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, -3800.0], N[(R$95$m * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3800

   1. Initial program 50.2%

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

      1. hypot-def 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in phi2 around 0 45.9%

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

      1. +-commutative 45.9%

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

      2. unpow2 45.9%

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

      3. unpow2 45.9%

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

      4. unpow2 45.9%

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

      5. unswap-sqr 45.9%

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

      6. hypot-def 79.1%

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

   7. Simplified 79.1%

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

   if -3800 < phi1

   1. Initial program 63.6%

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

      1. hypot-def 97.6%

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

   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 53.5%

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

      1. +-commutative 53.5%

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

      2. unpow2 53.5%

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

      3. unpow2 53.5%

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

      4. unpow2 53.5%

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

      5. unswap-sqr 53.5%

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

      6. hypot-def 80.6%

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

   7. Simplified 80.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% 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 1 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 60.2%

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

   1. hypot-def 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. Final simplification 96.2%

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

Alternative 7: 71.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{+61}:\\ \;\;\;\;R_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 4.8e+61)
    (* R_m (hypot (- lambda1 lambda2) phi1))
    (* R_m (- phi2 phi1)))))

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 <= 4.8e+61) {
		tmp = R_m * hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R_m * (phi2 - phi1);
	}
	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 (phi2 <= 4.8e+61) {
		tmp = R_m * Math.hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R_m * (phi2 - phi1);
	}
	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 <= 4.8e+61:
		tmp = R_m * math.hypot((lambda1 - lambda2), phi1)
	else:
		tmp = R_m * (phi2 - phi1)
	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 <= 4.8e+61)
		tmp = Float64(R_m * hypot(Float64(lambda1 - lambda2), phi1));
	else
		tmp = Float64(R_m * Float64(phi2 - phi1));
	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 <= 4.8e+61)
		tmp = R_m * hypot((lambda1 - lambda2), phi1);
	else
		tmp = R_m * (phi2 - phi1);
	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, 4.8e+61], N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.7999999999999998e61

   1. Initial program 65.9%

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

      1. hypot-def 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi1 around 0 92.0%

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

   6. Taylor expanded in phi2 around 0 55.9%

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

      1. +-commutative 55.9%

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

      2. unpow2 55.9%

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

      3. unpow2 55.9%

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

      4. hypot-def 75.8%

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

   8. Simplified 75.8%

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

   if 4.7999999999999998e61 < phi2

   1. Initial program 36.1%

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

      1. hypot-def 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in phi1 around -inf 70.8%

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

      1. +-commutative 70.8%

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

      2. mul-1-neg 70.8%

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

      3. unsub-neg 70.8%

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

      4. *-commutative 70.8%

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

      5. *-commutative 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in R around -inf 70.8%

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

      1. associate-*r* 70.8%

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

      2. neg-mul-1 70.8%

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

      3. distribute-lft-out-- 70.8%

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

      4. neg-mul-1 70.8%

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

      5. distribute-lft-neg-in 70.8%

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

      6. neg-mul-1 70.8%

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

      7. distribute-lft-out-- 70.8%

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

      8. cancel-sign-sub-inv 70.8%

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

      9. metadata-eval 70.8%

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

      10. *-lft-identity 70.8%

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

      11. distribute-lft-in 70.8%

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

      12. mul-1-neg 70.8%

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

      13. distribute-rgt-neg-in 70.8%

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

      14. mul-1-neg 70.8%

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

      15. +-commutative 70.8%

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

      16. mul-1-neg 70.8%

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

      17. distribute-rgt-neg-in 70.8%

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

      18. mul-1-neg 70.8%

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

      19. distribute-lft-in 70.8%

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

      20. mul-1-neg 70.8%

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

      21. sub-neg 70.8%

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

      22. *-commutative 70.8%

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

   10. Simplified 70.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{+61}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.2% accurate, 36.5× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0078:\\ \;\;\;\;R_m \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi2 0.0078) (* 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 (phi2 <= 0.0078) {
		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 (phi2 <= 0.0078d0) 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 (phi2 <= 0.0078) {
		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 phi2 <= 0.0078:
		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 (phi2 <= 0.0078)
		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 (phi2 <= 0.0078)
		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[phi2, 0.0078], N[(R$95$m * (-phi1)), $MachinePrecision], N[(R$95$m * phi2), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0077999999999999996

   1. Initial program 65.7%

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

      1. hypot-def 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in phi1 around -inf 23.9%

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

      1. mul-1-neg 23.9%

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

      2. *-commutative 23.9%

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

      3. distribute-rgt-neg-in 23.9%

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

   7. Simplified 23.9%

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

   if 0.0077999999999999996 < phi2

   1. Initial program 43.6%

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

      1. hypot-def 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in phi2 around inf 61.6%

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

      1. *-commutative 61.6%

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

   7. Simplified 61.6%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0078:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.3% accurate, 65.8× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ R_s \cdot \left(R_m \cdot \left(\phi_2 - \phi_1\right)\right) \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (- phi2 phi1))))

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 - phi1));
}

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 - phi1))
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 - phi1));
}

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 - phi1))

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 * Float64(phi2 - phi1)))
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 - phi1));
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[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

   1. hypot-def 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 phi1 around -inf 33.7%

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

   1. +-commutative 33.7%

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

   2. mul-1-neg 33.7%

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

   3. unsub-neg 33.7%

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

   4. *-commutative 33.7%

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

   5. *-commutative 33.7%

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

7. Simplified 33.7%

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

8. Taylor expanded in R around -inf 34.5%

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

   1. associate-*r* 34.5%

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

   2. neg-mul-1 34.5%

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

   3. distribute-lft-out-- 34.5%

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

   4. neg-mul-1 34.5%

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

   5. distribute-lft-neg-in 34.5%

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

   6. neg-mul-1 34.5%

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

   7. distribute-lft-out-- 34.5%

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

   8. cancel-sign-sub-inv 34.5%

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

   9. metadata-eval 34.5%

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

   10. *-lft-identity 34.5%

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

   11. distribute-lft-in 33.7%

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

   12. mul-1-neg 33.7%

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

   13. distribute-rgt-neg-in 33.7%

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

   14. mul-1-neg 33.7%

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

   15. +-commutative 33.7%

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

   16. mul-1-neg 33.7%

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

   17. distribute-rgt-neg-in 33.7%

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

   18. mul-1-neg 33.7%

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

   19. distribute-lft-in 34.5%

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

   20. mul-1-neg 34.5%

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

   21. sub-neg 34.5%

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

   22. *-commutative 34.5%

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

10. Simplified 34.5%

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

11. Final simplification 34.5%

    \[\leadsto R \cdot \left(\phi_2 - \phi_1\right) \]
12. Add Preprocessing

Alternative 10: 16.9% 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 1 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 60.2%

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

   1. hypot-def 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 inf 19.4%

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

   1. *-commutative 19.4%

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

7. Simplified 19.4%

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

8. Final simplification 19.4%

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

Reproduce

herbie shell --seed 2024024 
(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))))))