VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.2%

Time: 13.4s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = -2.922833550873919e+69
  2. l = -3470104525439398.5

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: 76.6% 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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 2000000000000:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

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) 2000000000000.0)
    (- (* PI l_m) (/ (/ (tan (* 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) <= 2000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / 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) <= 2000000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / 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) <= 2000000000000.0:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / 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) <= 2000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / 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) <= 2000000000000.0)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / 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], 2000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e12

   1. Initial program 80.6%

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

      1. associate-*l/ 81.0%

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

      2. *-un-lft-identity 81.0%

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

      3. associate-/r* 84.8%

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

   3. Applied egg-rr 84.8%

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

   if 2e12 < (*.f64 (PI.f64) l)

   1. Initial program 57.9%

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

      1. sqr-neg 57.9%

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

      2. associate-*l/ 57.9%

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

      3. sqr-neg 57.9%

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

      4. *-lft-identity 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in l around 0 41.9%

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

   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 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 7.2 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{elif}\;\pi \cdot l_m \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{l_m \cdot \left(-\pi\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot l_m \leq 10^{-94} \lor \neg \left(\pi \cdot l_m \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l_m}{{F}^{2}} \cdot \left(-\pi\right)\\ \end{array} \end{array} \]

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) 7.2e-241)
    (* PI l_m)
    (if (<= (* PI l_m) 5e-136)
      (/ (/ (* l_m (- PI)) F) F)
      (if (or (<= (* PI l_m) 1e-94) (not (<= (* PI l_m) 2e-61)))
        (* PI l_m)
        (* (/ l_m (pow F 2.0)) (- PI)))))))

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) <= 7.2e-241) {
		tmp = ((double) M_PI) * l_m;
	} else if ((((double) M_PI) * l_m) <= 5e-136) {
		tmp = ((l_m * -((double) M_PI)) / F) / F;
	} else if (((((double) M_PI) * l_m) <= 1e-94) || !((((double) M_PI) * l_m) <= 2e-61)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = (l_m / pow(F, 2.0)) * -((double) M_PI);
	}
	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) <= 7.2e-241) {
		tmp = Math.PI * l_m;
	} else if ((Math.PI * l_m) <= 5e-136) {
		tmp = ((l_m * -Math.PI) / F) / F;
	} else if (((Math.PI * l_m) <= 1e-94) || !((Math.PI * l_m) <= 2e-61)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = (l_m / Math.pow(F, 2.0)) * -Math.PI;
	}
	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) <= 7.2e-241:
		tmp = math.pi * l_m
	elif (math.pi * l_m) <= 5e-136:
		tmp = ((l_m * -math.pi) / F) / F
	elif ((math.pi * l_m) <= 1e-94) or not ((math.pi * l_m) <= 2e-61):
		tmp = math.pi * l_m
	else:
		tmp = (l_m / math.pow(F, 2.0)) * -math.pi
	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) <= 7.2e-241)
		tmp = Float64(pi * l_m);
	elseif (Float64(pi * l_m) <= 5e-136)
		tmp = Float64(Float64(Float64(l_m * Float64(-pi)) / F) / F);
	elseif ((Float64(pi * l_m) <= 1e-94) || !(Float64(pi * l_m) <= 2e-61))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(l_m / (F ^ 2.0)) * Float64(-pi));
	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) <= 7.2e-241)
		tmp = pi * l_m;
	elseif ((pi * l_m) <= 5e-136)
		tmp = ((l_m * -pi) / F) / F;
	elseif (((pi * l_m) <= 1e-94) || ~(((pi * l_m) <= 2e-61)))
		tmp = pi * l_m;
	else
		tmp = (l_m / (F ^ 2.0)) * -pi;
	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], 7.2e-241], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e-136], N[(N[(N[(l$95$m * (-Pi)), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision], If[Or[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-94], N[Not[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-61]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(l$95$m / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] * (-Pi)), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 7.1999999999999998e-241 or 5.0000000000000002e-136 < (*.f64 (PI.f64) l) < 9.9999999999999996e-95 or 2.0000000000000001e-61 < (*.f64 (PI.f64) l)

   1. Initial program 74.0%

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

      1. sqr-neg 74.0%

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

      2. associate-*l/ 74.0%

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

      3. sqr-neg 74.0%

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

      4. *-lft-identity 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in l around 0 63.9%

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

   5. Taylor expanded in F around inf 80.9%

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

   if 7.1999999999999998e-241 < (*.f64 (PI.f64) l) < 5.0000000000000002e-136

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 94.5%

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

      3. sqr-neg 94.5%

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

      4. *-lft-identity 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in l around 0 94.5%

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

   5. Taylor expanded in F around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. associate-*l/ 57.9%

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

      3. *-commutative 57.9%

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

   7. Simplified 57.9%

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

      1. associate-*r/ 57.8%

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

      2. unpow2 57.8%

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

      3. associate-/r* 62.9%

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

   9. Applied egg-rr 62.9%

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

   if 9.9999999999999996e-95 < (*.f64 (PI.f64) l) < 2.0000000000000001e-61

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. associate-*l/ 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 99.2%

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

   5. Taylor expanded in F around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-*l/ 75.5%

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

      3. *-commutative 75.5%

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

   7. Simplified 75.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 7.2 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(-\pi\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-94} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{F}^{2}} \cdot \left(-\pi\right)\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 7.2 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{elif}\;\pi \cdot l_m \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{l_m \cdot \left(-\pi\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot l_m \leq 10^{-94} \lor \neg \left(\pi \cdot l_m \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{l_m}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \end{array} \]

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) 7.2e-241)
    (* PI l_m)
    (if (<= (* PI l_m) 5e-136)
      (/ (/ (* l_m (- PI)) F) F)
      (if (or (<= (* PI l_m) 1e-94) (not (<= (* PI l_m) 2e-61)))
        (* PI l_m)
        (/ (* (/ l_m F) (- PI)) F))))))

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) <= 7.2e-241) {
		tmp = ((double) M_PI) * l_m;
	} else if ((((double) M_PI) * l_m) <= 5e-136) {
		tmp = ((l_m * -((double) M_PI)) / F) / F;
	} else if (((((double) M_PI) * l_m) <= 1e-94) || !((((double) M_PI) * l_m) <= 2e-61)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = ((l_m / F) * -((double) M_PI)) / F;
	}
	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) <= 7.2e-241) {
		tmp = Math.PI * l_m;
	} else if ((Math.PI * l_m) <= 5e-136) {
		tmp = ((l_m * -Math.PI) / F) / F;
	} else if (((Math.PI * l_m) <= 1e-94) || !((Math.PI * l_m) <= 2e-61)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = ((l_m / F) * -Math.PI) / F;
	}
	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) <= 7.2e-241:
		tmp = math.pi * l_m
	elif (math.pi * l_m) <= 5e-136:
		tmp = ((l_m * -math.pi) / F) / F
	elif ((math.pi * l_m) <= 1e-94) or not ((math.pi * l_m) <= 2e-61):
		tmp = math.pi * l_m
	else:
		tmp = ((l_m / F) * -math.pi) / F
	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) <= 7.2e-241)
		tmp = Float64(pi * l_m);
	elseif (Float64(pi * l_m) <= 5e-136)
		tmp = Float64(Float64(Float64(l_m * Float64(-pi)) / F) / F);
	elseif ((Float64(pi * l_m) <= 1e-94) || !(Float64(pi * l_m) <= 2e-61))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(Float64(l_m / F) * Float64(-pi)) / F);
	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) <= 7.2e-241)
		tmp = pi * l_m;
	elseif ((pi * l_m) <= 5e-136)
		tmp = ((l_m * -pi) / F) / F;
	elseif (((pi * l_m) <= 1e-94) || ~(((pi * l_m) <= 2e-61)))
		tmp = pi * l_m;
	else
		tmp = ((l_m / F) * -pi) / F;
	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], 7.2e-241], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e-136], N[(N[(N[(l$95$m * (-Pi)), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision], If[Or[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-94], N[Not[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-61]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(N[(l$95$m / F), $MachinePrecision] * (-Pi)), $MachinePrecision] / F), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 7.1999999999999998e-241 or 5.0000000000000002e-136 < (*.f64 (PI.f64) l) < 9.9999999999999996e-95 or 2.0000000000000001e-61 < (*.f64 (PI.f64) l)

   1. Initial program 74.0%

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

      1. sqr-neg 74.0%

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

      2. associate-*l/ 74.0%

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

      3. sqr-neg 74.0%

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

      4. *-lft-identity 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in l around 0 63.9%

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

   5. Taylor expanded in F around inf 80.9%

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

   if 7.1999999999999998e-241 < (*.f64 (PI.f64) l) < 5.0000000000000002e-136

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 94.5%

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

      3. sqr-neg 94.5%

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

      4. *-lft-identity 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in l around 0 94.5%

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

   5. Taylor expanded in F around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. associate-*l/ 57.9%

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

      3. *-commutative 57.9%

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

   7. Simplified 57.9%

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

      1. associate-*r/ 57.8%

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

      2. unpow2 57.8%

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

      3. associate-/r* 62.9%

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

   9. Applied egg-rr 62.9%

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

   if 9.9999999999999996e-95 < (*.f64 (PI.f64) l) < 2.0000000000000001e-61

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. associate-*l/ 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 99.2%

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

   5. Taylor expanded in F around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-*l/ 75.5%

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

      3. *-commutative 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*r/ 75.5%

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

      2. unpow2 75.5%

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

      3. frac-times 75.9%

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

      4. associate-*l/ 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 7.2 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(-\pi\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-94} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \]

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 8 \cdot 10^{-15}:\\ \;\;\;\;\pi \cdot l_m - \frac{\pi}{F} \cdot \frac{l_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

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) 8e-15) (- (* PI l_m) (* (/ PI F) (/ l_m 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) <= 8e-15) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / 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) <= 8e-15) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / 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) <= 8e-15:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / 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) <= 8e-15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / 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) <= 8e-15)
		tmp = (pi * l_m) - ((pi / F) * (l_m / 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], 8e-15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 8.0000000000000006e-15

   1. Initial program 79.8%

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

      1. sqr-neg 79.8%

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

      2. associate-*l/ 80.2%

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

      3. sqr-neg 80.2%

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

      4. *-lft-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in l around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

   if 8.0000000000000006e-15 < (*.f64 (PI.f64) l)

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. sqr-neg 63.2%

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

      4. *-lft-identity 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 8 \cdot 10^{-15}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 8 \cdot 10^{-15}:\\ \;\;\;\;\left(\pi \cdot l_m\right) \cdot \left(1 - {F}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

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) 8e-15) (* (* PI l_m) (- 1.0 (pow F -2.0))) (* 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) <= 8e-15) {
		tmp = (((double) M_PI) * l_m) * (1.0 - pow(F, -2.0));
	} 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) <= 8e-15) {
		tmp = (Math.PI * l_m) * (1.0 - Math.pow(F, -2.0));
	} 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) <= 8e-15:
		tmp = (math.pi * l_m) * (1.0 - math.pow(F, -2.0))
	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) <= 8e-15)
		tmp = Float64(Float64(pi * l_m) * Float64(1.0 - (F ^ -2.0)));
	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) <= 8e-15)
		tmp = (pi * l_m) * (1.0 - (F ^ -2.0));
	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], 8e-15], N[(N[(Pi * l$95$m), $MachinePrecision] * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 8.0000000000000006e-15

   1. Initial program 79.8%

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

      1. sqr-neg 79.8%

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

      2. associate-*l/ 80.2%

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

      3. sqr-neg 80.2%

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

      4. *-lft-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in l around 0 73.3%

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

   5. Taylor expanded in l around 0 73.3%

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

      1. *-lft-identity 73.3%

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

      2. associate-*l/ 73.3%

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

      3. unpow2 73.3%

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

      4. associate-/r* 73.4%

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

      5. *-rgt-identity 73.4%

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

      6. associate-*r/ 73.3%

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

      7. unpow-1 73.3%

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

      8. unpow-1 73.3%

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

      9. pow-sqr 73.4%

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

      10. metadata-eval 73.4%

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

      11. *-commutative 73.4%

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

      12. distribute-lft-out-- 73.4%

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

      13. *-commutative 73.4%

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

      14. associate-*r* 72.9%

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

      15. *-commutative 72.9%

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

      16. *-rgt-identity 72.9%

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

      17. distribute-lft-out-- 73.4%

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

   7. Simplified 73.4%

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

   if 8.0000000000000006e-15 < (*.f64 (PI.f64) l)

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. sqr-neg 63.2%

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

      4. *-lft-identity 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 8 \cdot 10^{-15}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6: 74.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.26 \cdot 10^{-241} \lor \neg \left(l_m \leq 7.5 \cdot 10^{-136} \lor \neg \left(l_m \leq 4.2 \cdot 10^{-93}\right) \land l_m \leq 9 \cdot 10^{-60}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{l_m}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= l_m 2.26e-241)
          (not
           (or (<= l_m 7.5e-136) (and (not (<= l_m 4.2e-93)) (<= l_m 9e-60)))))
    (* PI l_m)
    (/ (* (/ l_m F) (- PI)) F))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 2.26e-241) || !((l_m <= 7.5e-136) || (!(l_m <= 4.2e-93) && (l_m <= 9e-60)))) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = ((l_m / F) * -((double) M_PI)) / F;
	}
	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 ((l_m <= 2.26e-241) || !((l_m <= 7.5e-136) || (!(l_m <= 4.2e-93) && (l_m <= 9e-60)))) {
		tmp = Math.PI * l_m;
	} else {
		tmp = ((l_m / F) * -Math.PI) / F;
	}
	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 (l_m <= 2.26e-241) or not ((l_m <= 7.5e-136) or (not (l_m <= 4.2e-93) and (l_m <= 9e-60))):
		tmp = math.pi * l_m
	else:
		tmp = ((l_m / F) * -math.pi) / F
	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 ((l_m <= 2.26e-241) || !((l_m <= 7.5e-136) || (!(l_m <= 4.2e-93) && (l_m <= 9e-60))))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(Float64(l_m / F) * Float64(-pi)) / F);
	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 ((l_m <= 2.26e-241) || ~(((l_m <= 7.5e-136) || (~((l_m <= 4.2e-93)) && (l_m <= 9e-60)))))
		tmp = pi * l_m;
	else
		tmp = ((l_m / F) * -pi) / F;
	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[l$95$m, 2.26e-241], N[Not[Or[LessEqual[l$95$m, 7.5e-136], And[N[Not[LessEqual[l$95$m, 4.2e-93]], $MachinePrecision], LessEqual[l$95$m, 9e-60]]]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(N[(l$95$m / F), $MachinePrecision] * (-Pi)), $MachinePrecision] / F), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.26e-241 or 7.5000000000000003e-136 < l < 4.2000000000000002e-93 or 9.00000000000000001e-60 < l

   1. Initial program 74.0%

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

      1. sqr-neg 74.0%

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

      2. associate-*l/ 74.0%

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

      3. sqr-neg 74.0%

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

      4. *-lft-identity 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in l around 0 63.9%

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

   5. Taylor expanded in F around inf 80.9%

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

   if 2.26e-241 < l < 7.5000000000000003e-136 or 4.2000000000000002e-93 < l < 9.00000000000000001e-60

   1. Initial program 91.9%

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

      1. sqr-neg 91.9%

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

      2. associate-*l/ 95.3%

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

      3. sqr-neg 95.3%

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

      4. *-lft-identity 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in l around 0 95.3%

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

   5. Taylor expanded in F around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. associate-*l/ 60.7%

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

      3. *-commutative 60.7%

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

   7. Simplified 60.7%

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

      1. associate-*r/ 60.6%

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

      2. unpow2 60.6%

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

      3. frac-times 65.1%

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

      4. associate-*l/ 64.9%

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

   9. Applied egg-rr 64.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.26 \cdot 10^{-241} \lor \neg \left(\ell \leq 7.5 \cdot 10^{-136} \lor \neg \left(\ell \leq 4.2 \cdot 10^{-93}\right) \land \ell \leq 9 \cdot 10^{-60}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \]

Alternative 7: 74.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.26 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{elif}\;l_m \leq 2.45 \cdot 10^{-136}:\\ \;\;\;\;\frac{l_m \cdot \frac{-\pi}{F}}{F}\\ \mathbf{elif}\;l_m \leq 2.55 \cdot 10^{-89} \lor \neg \left(l_m \leq 2.9 \cdot 10^{-54}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{l_m}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 2.26e-241)
    (* PI l_m)
    (if (<= l_m 2.45e-136)
      (/ (* l_m (/ (- PI) F)) F)
      (if (or (<= l_m 2.55e-89) (not (<= l_m 2.9e-54)))
        (* PI l_m)
        (/ (* (/ l_m F) (- PI)) F))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 2.26e-241) {
		tmp = ((double) M_PI) * l_m;
	} else if (l_m <= 2.45e-136) {
		tmp = (l_m * (-((double) M_PI) / F)) / F;
	} else if ((l_m <= 2.55e-89) || !(l_m <= 2.9e-54)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = ((l_m / F) * -((double) M_PI)) / F;
	}
	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 (l_m <= 2.26e-241) {
		tmp = Math.PI * l_m;
	} else if (l_m <= 2.45e-136) {
		tmp = (l_m * (-Math.PI / F)) / F;
	} else if ((l_m <= 2.55e-89) || !(l_m <= 2.9e-54)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = ((l_m / F) * -Math.PI) / F;
	}
	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 l_m <= 2.26e-241:
		tmp = math.pi * l_m
	elif l_m <= 2.45e-136:
		tmp = (l_m * (-math.pi / F)) / F
	elif (l_m <= 2.55e-89) or not (l_m <= 2.9e-54):
		tmp = math.pi * l_m
	else:
		tmp = ((l_m / F) * -math.pi) / F
	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 (l_m <= 2.26e-241)
		tmp = Float64(pi * l_m);
	elseif (l_m <= 2.45e-136)
		tmp = Float64(Float64(l_m * Float64(Float64(-pi) / F)) / F);
	elseif ((l_m <= 2.55e-89) || !(l_m <= 2.9e-54))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(Float64(l_m / F) * Float64(-pi)) / F);
	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 (l_m <= 2.26e-241)
		tmp = pi * l_m;
	elseif (l_m <= 2.45e-136)
		tmp = (l_m * (-pi / F)) / F;
	elseif ((l_m <= 2.55e-89) || ~((l_m <= 2.9e-54)))
		tmp = pi * l_m;
	else
		tmp = ((l_m / F) * -pi) / F;
	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[l$95$m, 2.26e-241], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[l$95$m, 2.45e-136], N[(N[(l$95$m * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision], If[Or[LessEqual[l$95$m, 2.55e-89], N[Not[LessEqual[l$95$m, 2.9e-54]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(N[(l$95$m / F), $MachinePrecision] * (-Pi)), $MachinePrecision] / F), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 2.26e-241 or 2.45e-136 < l < 2.55000000000000002e-89 or 2.90000000000000015e-54 < l

   1. Initial program 74.0%

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

      1. sqr-neg 74.0%

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

      2. associate-*l/ 74.0%

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

      3. sqr-neg 74.0%

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

      4. *-lft-identity 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in l around 0 63.9%

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

   5. Taylor expanded in F around inf 80.9%

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

   if 2.26e-241 < l < 2.45e-136

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 94.5%

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

      3. sqr-neg 94.5%

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

      4. *-lft-identity 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in l around 0 94.5%

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

   5. Taylor expanded in F around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. associate-*l/ 57.9%

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

      3. *-commutative 57.9%

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

   7. Simplified 57.9%

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

      1. associate-*r/ 57.8%

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

      2. unpow2 57.8%

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

      3. frac-times 63.0%

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

      4. associate-*r/ 63.1%

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

   9. Applied egg-rr 63.1%

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

   if 2.55000000000000002e-89 < l < 2.90000000000000015e-54

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. associate-*l/ 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 99.2%

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

   5. Taylor expanded in F around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-*l/ 75.5%

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

      3. *-commutative 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*r/ 75.5%

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

      2. unpow2 75.5%

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

      3. frac-times 75.9%

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

      4. associate-*l/ 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.26 \cdot 10^{-241}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-136}:\\ \;\;\;\;\frac{\ell \cdot \frac{-\pi}{F}}{F}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-89} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-54}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{F} \cdot \left(-\pi\right)}{F}\\ \end{array} \]

Alternative 8: 74.0% 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 75.7%

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

   1. sqr-neg 75.7%

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

   2. associate-*l/ 76.0%

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

   3. sqr-neg 76.0%

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

   4. *-lft-identity 76.0%

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

3. Simplified 76.0%

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

4. Taylor expanded in l around 0 67.0%

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

5. Taylor expanded in F around inf 76.2%

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

6. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))