VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.0% → 97.7%

Time: 12.6s

Alternatives: 5

Speedup: 1.0×

• could not determine a ground truth

  1. F = 2.192035385762351e+138
  2. l = -3557522065159502.5

Specification

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{-\mathsf{fma}\left({l_m}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \pi \cdot \frac{l_m}{F}\right)}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-6)
    (fma
     PI
     l_m
     (/
      (-
       (fma
        (pow l_m 3.0)
        (* (/ (pow PI 3.0) F) 0.3333333333333333)
        (* PI (/ l_m F))))
      F))
    (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e-6) {
		tmp = fma(((double) M_PI), l_m, (-fma(pow(l_m, 3.0), ((pow(((double) M_PI), 3.0) / F) * 0.3333333333333333), (((double) M_PI) * (l_m / F))) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e-6)
		tmp = fma(pi, l_m, Float64(Float64(-fma((l_m ^ 3.0), Float64(Float64((pi ^ 3.0) / F) * 0.3333333333333333), Float64(pi * Float64(l_m / F)))) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-6], N[(Pi * l$95$m + N[((-N[(N[Power[l$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / F), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.99999999999999991e-6

   1. Initial program 81.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. fma-neg 81.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \ell, -\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\right)} \]

      2. distribute-lft-neg-in 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \color{blue}{\left(-\frac{1}{F \cdot F}\right) \cdot \tan \left(\pi \cdot \ell\right)}\right) \]

      3. sqr-neg 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \left(-\frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}}\right) \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      4. distribute-neg-frac 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{-1}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      5. metadata-eval 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-1}}{\left(-F\right) \cdot \left(-F\right)} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      6. distribute-lft-neg-out 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-1}{\color{blue}{-F \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      7. neg-mul-1 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-1}{\color{blue}{-1 \cdot \left(F \cdot \left(-F\right)\right)}} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      8. associate-/r* 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\frac{-1}{-1}}{F \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      9. metadata-eval 81.2%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{1}}{F \cdot \left(-F\right)} \cdot \tan \left(\pi \cdot \ell\right)\right) \]

      10. associate-*l/ 81.2%

          \[\leadsto \mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot \left(-F\right)}}\right) \]

      11. *-lft-identity 81.2%

          \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot \left(-F\right)}\right) \]

      12. associate-/l/ 87.7%

          \[\leadsto \mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}}\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)} \]

   4. Taylor expanded in l around 0 72.8%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}\right)\right) + -1 \cdot \frac{\ell \cdot \pi}{F}}}{F}\right) \]
   5. Step-by-step derivation

      1. distribute-lft-out 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}\right) + \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      2. mul-1-neg 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-\left({\ell}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}\right) + \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      3. fma-def 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\color{blue}{\mathsf{fma}\left({\ell}^{3}, -0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}, \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      4. distribute-rgt-out-- 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \color{blue}{\frac{{\pi}^{3}}{F} \cdot \left(-0.16666666666666666 - -0.5\right)}, \frac{\ell \cdot \pi}{F}\right)}{F}\right) \]

      5. metadata-eval 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot \color{blue}{0.3333333333333333}, \frac{\ell \cdot \pi}{F}\right)}{F}\right) \]

      6. associate-/l* 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \color{blue}{\frac{\ell}{\frac{F}{\pi}}}\right)}{F}\right) \]

      7. associate-/r/ 72.8%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \color{blue}{\frac{\ell}{F} \cdot \pi}\right)}{F}\right) \]

   6. Simplified 72.8%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \frac{\ell}{F} \cdot \pi\right)}}{F}\right) \]

   if 1.99999999999999991e-6 < (*.f64 (PI.f64) l)

   1. Initial program 67.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 67.1%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 67.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 67.1%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 67.1%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 54.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around inf 99.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \pi \cdot \frac{\ell}{F}\right)}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\pi \cdot l_m - \frac{l_m}{F} \cdot \frac{\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-6) (- (* PI l_m) (* (/ l_m F) (/ PI F))) (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e-6) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 2e-6) {
		tmp = (Math.PI * l_m) - ((l_m / F) * (Math.PI / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 2e-6:
		tmp = (math.pi * l_m) - ((l_m / F) * (math.pi / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e-6)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e-6)
		tmp = (pi * l_m) - ((l_m / F) * (pi / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-6], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.99999999999999991e-6

   1. Initial program 81.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 81.2%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 81.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 81.2%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 81.2%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 81.2%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 76.2%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   5. Step-by-step derivation

      1. *-commutative 76.2%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. times-frac 82.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   6. Applied egg-rr 82.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   if 1.99999999999999991e-6 < (*.f64 (PI.f64) l)

   1. Initial program 67.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 67.1%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 67.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 67.1%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 67.1%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 54.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around inf 99.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3: 50.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \frac{l_m}{F} \cdot \frac{-\pi}{F}\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{-222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-71}:\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{elif}\;F \leq 5.1 \cdot 10^{-53}:\\ \;\;\;\;\left(l_m \cdot {F}^{-2}\right) \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-19} \lor \neg \left(F \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (/ l_m F) (/ (- PI) F))))
   (*
    l_s
    (if (<= F 8e-222)
      t_0
      (if (<= F 1.36e-71)
        (* PI l_m)
        (if (<= F 5.1e-53)
          (* (* l_m (pow F -2.0)) (- PI))
          (if (or (<= F 1.22e-19) (not (<= F 4.8e-9))) (* PI l_m) t_0)))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = (l_m / F) * (-((double) M_PI) / F);
	double tmp;
	if (F <= 8e-222) {
		tmp = t_0;
	} else if (F <= 1.36e-71) {
		tmp = ((double) M_PI) * l_m;
	} else if (F <= 5.1e-53) {
		tmp = (l_m * pow(F, -2.0)) * -((double) M_PI);
	} else if ((F <= 1.22e-19) || !(F <= 4.8e-9)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = t_0;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = (l_m / F) * (-Math.PI / F);
	double tmp;
	if (F <= 8e-222) {
		tmp = t_0;
	} else if (F <= 1.36e-71) {
		tmp = Math.PI * l_m;
	} else if (F <= 5.1e-53) {
		tmp = (l_m * Math.pow(F, -2.0)) * -Math.PI;
	} else if ((F <= 1.22e-19) || !(F <= 4.8e-9)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = t_0;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = (l_m / F) * (-math.pi / F)
	tmp = 0
	if F <= 8e-222:
		tmp = t_0
	elif F <= 1.36e-71:
		tmp = math.pi * l_m
	elif F <= 5.1e-53:
		tmp = (l_m * math.pow(F, -2.0)) * -math.pi
	elif (F <= 1.22e-19) or not (F <= 4.8e-9):
		tmp = math.pi * l_m
	else:
		tmp = t_0
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(l_m / F) * Float64(Float64(-pi) / F))
	tmp = 0.0
	if (F <= 8e-222)
		tmp = t_0;
	elseif (F <= 1.36e-71)
		tmp = Float64(pi * l_m);
	elseif (F <= 5.1e-53)
		tmp = Float64(Float64(l_m * (F ^ -2.0)) * Float64(-pi));
	elseif ((F <= 1.22e-19) || !(F <= 4.8e-9))
		tmp = Float64(pi * l_m);
	else
		tmp = t_0;
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	t_0 = (l_m / F) * (-pi / F);
	tmp = 0.0;
	if (F <= 8e-222)
		tmp = t_0;
	elseif (F <= 1.36e-71)
		tmp = pi * l_m;
	elseif (F <= 5.1e-53)
		tmp = (l_m * (F ^ -2.0)) * -pi;
	elseif ((F <= 1.22e-19) || ~((F <= 4.8e-9)))
		tmp = pi * l_m;
	else
		tmp = t_0;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[(N[(l$95$m / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[F, 8e-222], t$95$0, If[LessEqual[F, 1.36e-71], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[F, 5.1e-53], N[(N[(l$95$m * N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision] * (-Pi)), $MachinePrecision], If[Or[LessEqual[F, 1.22e-19], N[Not[LessEqual[F, 4.8e-9]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], t$95$0]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if F < 8.00000000000000038e-222 or 1.2200000000000001e-19 < F < 4.8e-9

   1. Initial program 75.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 75.6%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 75.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 75.7%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 75.7%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 70.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around 0 25.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 25.9%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

   7. Simplified 25.9%

      \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto -\frac{\color{blue}{\pi \cdot \ell}}{{F}^{2}} \]

      2. unpow2 25.9%

         \[\leadsto -\frac{\pi \cdot \ell}{\color{blue}{F \cdot F}} \]

      3. times-frac 29.7%

         \[\leadsto -\color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   9. Applied egg-rr 29.7%

      \[\leadsto -\color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   if 8.00000000000000038e-222 < F < 1.3599999999999999e-71 or 5.10000000000000045e-53 < F < 1.2200000000000001e-19 or 4.8e-9 < F

   1. Initial program 79.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 79.6%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 79.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 79.6%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 79.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 74.0%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around inf 83.1%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if 1.3599999999999999e-71 < F < 5.10000000000000045e-53

   1. Initial program 99.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 99.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 99.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 99.2%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 99.2%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 59.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around 0 59.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 59.4%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

   7. Simplified 59.4%

      \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]
   8. Step-by-step derivation

      1. add-log-exp 2.3%

         \[\leadsto -\color{blue}{\log \left(e^{\frac{\ell \cdot \pi}{{F}^{2}}}\right)} \]

      2. *-un-lft-identity 2.3%

         \[\leadsto -\log \color{blue}{\left(1 \cdot e^{\frac{\ell \cdot \pi}{{F}^{2}}}\right)} \]

      3. log-prod 2.3%

         \[\leadsto -\color{blue}{\left(\log 1 + \log \left(e^{\frac{\ell \cdot \pi}{{F}^{2}}}\right)\right)} \]

      4. metadata-eval 2.3%

         \[\leadsto -\left(\color{blue}{0} + \log \left(e^{\frac{\ell \cdot \pi}{{F}^{2}}}\right)\right) \]

      5. add-log-exp 59.4%

         \[\leadsto -\left(0 + \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}}\right) \]

      6. div-inv 59.7%

         \[\leadsto -\left(0 + \color{blue}{\left(\ell \cdot \pi\right) \cdot \frac{1}{{F}^{2}}}\right) \]

      7. associate-*l* 59.4%

         \[\leadsto -\left(0 + \color{blue}{\ell \cdot \left(\pi \cdot \frac{1}{{F}^{2}}\right)}\right) \]

      8. pow-flip 59.4%

         \[\leadsto -\left(0 + \ell \cdot \left(\pi \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right) \]

      9. metadata-eval 59.4%

         \[\leadsto -\left(0 + \ell \cdot \left(\pi \cdot {F}^{\color{blue}{-2}}\right)\right) \]

   9. Applied egg-rr 59.4%

      \[\leadsto -\color{blue}{\left(0 + \ell \cdot \left(\pi \cdot {F}^{-2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-lft-identity 59.4%

          \[\leadsto -\color{blue}{\ell \cdot \left(\pi \cdot {F}^{-2}\right)} \]

       2. *-commutative 59.4%

          \[\leadsto -\color{blue}{\left(\pi \cdot {F}^{-2}\right) \cdot \ell} \]

       3. associate-*r* 59.7%

          \[\leadsto -\color{blue}{\pi \cdot \left({F}^{-2} \cdot \ell\right)} \]

       4. *-commutative 59.7%

          \[\leadsto -\pi \cdot \color{blue}{\left(\ell \cdot {F}^{-2}\right)} \]

   11. Simplified 59.7%

       \[\leadsto -\color{blue}{\pi \cdot \left(\ell \cdot {F}^{-2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{-222}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-71}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 5.1 \cdot 10^{-53}:\\ \;\;\;\;\left(\ell \cdot {F}^{-2}\right) \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-19} \lor \neg \left(F \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \]

Alternative 4: 50.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{-221} \lor \neg \left(F \leq 3.6 \cdot 10^{-72}\right) \land \left(F \leq 5.15 \cdot 10^{-54} \lor \neg \left(F \leq 9.5 \cdot 10^{-20}\right) \land F \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{l_m}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= F 8e-221)
          (and (not (<= F 3.6e-72))
               (or (<= F 5.15e-54) (and (not (<= F 9.5e-20)) (<= F 6.5e-9)))))
    (* (/ l_m F) (/ (- PI) F))
    (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((F <= 8e-221) || (!(F <= 3.6e-72) && ((F <= 5.15e-54) || (!(F <= 9.5e-20) && (F <= 6.5e-9))))) {
		tmp = (l_m / F) * (-((double) M_PI) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((F <= 8e-221) || (!(F <= 3.6e-72) && ((F <= 5.15e-54) || (!(F <= 9.5e-20) && (F <= 6.5e-9))))) {
		tmp = (l_m / F) * (-Math.PI / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (F <= 8e-221) or (not (F <= 3.6e-72) and ((F <= 5.15e-54) or (not (F <= 9.5e-20) and (F <= 6.5e-9)))):
		tmp = (l_m / F) * (-math.pi / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((F <= 8e-221) || (!(F <= 3.6e-72) && ((F <= 5.15e-54) || (!(F <= 9.5e-20) && (F <= 6.5e-9)))))
		tmp = Float64(Float64(l_m / F) * Float64(Float64(-pi) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((F <= 8e-221) || (~((F <= 3.6e-72)) && ((F <= 5.15e-54) || (~((F <= 9.5e-20)) && (F <= 6.5e-9)))))
		tmp = (l_m / F) * (-pi / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[Or[LessEqual[F, 8e-221], And[N[Not[LessEqual[F, 3.6e-72]], $MachinePrecision], Or[LessEqual[F, 5.15e-54], And[N[Not[LessEqual[F, 9.5e-20]], $MachinePrecision], LessEqual[F, 6.5e-9]]]]], N[(N[(l$95$m / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 8.00000000000000014e-221 or 3.6e-72 < F < 5.15000000000000007e-54 or 9.5e-20 < F < 6.5000000000000003e-9

   1. Initial program 77.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 77.2%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 77.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 77.2%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 77.2%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 69.7%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around 0 28.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 28.1%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

   7. Simplified 28.1%

      \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 28.1%

         \[\leadsto -\frac{\color{blue}{\pi \cdot \ell}}{{F}^{2}} \]

      2. unpow2 28.1%

         \[\leadsto -\frac{\pi \cdot \ell}{\color{blue}{F \cdot F}} \]

      3. times-frac 31.7%

         \[\leadsto -\color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   9. Applied egg-rr 31.7%

      \[\leadsto -\color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   if 8.00000000000000014e-221 < F < 3.6e-72 or 5.15000000000000007e-54 < F < 9.5e-20 or 6.5000000000000003e-9 < F

   1. Initial program 79.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 79.6%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. associate-*l/ 79.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. sqr-neg 79.6%

         \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

      4. *-lft-identity 79.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   4. Taylor expanded in l around 0 74.0%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

   5. Taylor expanded in F around inf 83.1%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{-221} \lor \neg \left(F \leq 3.6 \cdot 10^{-72}\right) \land \left(F \leq 5.15 \cdot 10^{-54} \lor \neg \left(F \leq 9.5 \cdot 10^{-20}\right) \land F \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5: 73.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* PI l_m)))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * (((double) M_PI) * l_m);
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * (Math.PI * l_m);
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * (math.pi * l_m)

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(pi * l_m))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * (pi * l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. sqr-neg 78.2%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. associate-*l/ 78.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. sqr-neg 78.2%

      \[\leadsto \pi \cdot \ell - \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   4. *-lft-identity 78.2%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

3. Simplified 78.2%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

4. Taylor expanded in l around 0 71.4%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]

5. Taylor expanded in F around inf 75.3%

   \[\leadsto \color{blue}{\ell \cdot \pi} \]

6. Final simplification 75.3%

   \[\leadsto \pi \cdot \ell \]

Reproduce

herbie shell --seed 2023333 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))