VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.7% → 99.1%

Time: 16.3s

Alternatives: 10

Speedup: 3.7×

• could not determine a ground truth

  1. F = -2.9387121224535758e+209
  2. l = 2268112272077526.5

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: 76.7% 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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-50}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-50)
    (- (* PI l_m) (* (/ PI F) (/ l_m F)))
    (if (<= (* PI l_m) 20000000000000.0)
      (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* 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) <= 1e-50) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else if ((((double) M_PI) * l_m) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (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) <= 1e-50) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else if ((Math.PI * l_m) <= 20000000000000.0) {
		tmp = (Math.PI * l_m) + (Math.tan((Math.PI * l_m)) * (-1.0 / (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) <= 1e-50:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	elif (math.pi * l_m) <= 20000000000000.0:
		tmp = (math.pi * l_m) + (math.tan((math.pi * l_m)) * (-1.0 / (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) <= 1e-50)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	elseif (Float64(pi * l_m) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(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) <= 1e-50)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	elseif ((pi * l_m) <= 20000000000000.0)
		tmp = (pi * l_m) + (tan((pi * l_m)) * (-1.0 / (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], 1e-50], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 1.00000000000000001e-50

   1. Initial program 82.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.9

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

   5. Simplified 75.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.7

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

   7. Applied egg-rr 79.7%

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

   if 1.00000000000000001e-50 < (*.f64 (PI.f64) l) < 2e13

   1. Initial program 99.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-50}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\pi \cdot \left(-l\_m\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -1e-294)
    (/ (* 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) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = (((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) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = (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) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294:
		tmp = (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(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -1e-294)
		tmp = Float64(Float64(pi * Float64(-l_m)) / Float64(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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294)
		tmp = (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[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-294], N[(N[(Pi * (-l$95$m)), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.00000000000000002e-294

   1. Initial program 79.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left({F}^{2}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left({F}^{2}\right)}} \]

      9. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({F}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{F \cdot F}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} \]

      13. lower-neg.f64 25.2

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

   8. Simplified 25.2%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{F \cdot \left(\mathsf{neg}\left(F\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}}{F \cdot \left(\mathsf{neg}\left(F\right)\right)} \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \ell \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(F \cdot F\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\color{blue}{F \cdot F}\right)} \]

      5. frac-2neg N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{F \cdot F}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{F \cdot F}} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F \cdot F} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \ell}\right)}{F \cdot F} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \ell}\right)}{F \cdot F} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot F}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \ell}\right)}{F \cdot F} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}\right)}{F \cdot F} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{F \cdot F} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{F \cdot F} \]

      15. lower-neg.f64 25.2

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

   10. Applied egg-rr 25.2%

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

   if -1.00000000000000002e-294 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 79.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 77.5

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

   8. Simplified 77.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;l\_m \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -1e-294)
    (* l_m (/ (- PI) (* 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) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = l_m * (-((double) M_PI) / (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) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = l_m * (-Math.PI / (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) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294:
		tmp = l_m * (-math.pi / (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(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -1e-294)
		tmp = Float64(l_m * Float64(Float64(-pi) / Float64(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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294)
		tmp = l_m * (-pi / (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[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-294], N[(l$95$m * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.00000000000000002e-294

   1. Initial program 79.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left({F}^{2}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left({F}^{2}\right)}} \]

      9. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({F}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{F \cdot F}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} \]

      13. lower-neg.f64 25.2

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

   8. Simplified 25.2%

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

   if -1.00000000000000002e-294 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 79.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 77.5

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

   8. Simplified 77.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5e-57)
    (- (* PI l_m) (* (/ PI F) (/ l_m F)))
    (if (<= (* PI l_m) 20000000000000.0)
      (- (* 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) <= 5e-57) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else if ((((double) M_PI) * l_m) <= 20000000000000.0) {
		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) <= 5e-57) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else if ((Math.PI * l_m) <= 20000000000000.0) {
		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) <= 5e-57:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	elif (math.pi * l_m) <= 20000000000000.0:
		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) <= 5e-57)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	elseif (Float64(pi * l_m) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(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) <= 5e-57)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	elseif ((pi * l_m) <= 20000000000000.0)
		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], 5e-57], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 5.0000000000000002e-57

   1. Initial program 82.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.5

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

   7. Applied egg-rr 79.5%

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

   if 5.0000000000000002e-57 < (*.f64 (PI.f64) l) < 2e13

   1. Initial program 99.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot {F}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot {F}^{2}}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot {F}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot {F}^{2}} \]

      9. lower-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot {F}^{2}} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(F \cdot F\right)}} \]

      11. lower-*.f64 99.7

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

   5. Simplified 99.7%

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

      1. lift-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(F \cdot F\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(F \cdot F\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(F \cdot F\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(F \cdot F\right)}} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F \cdot F}} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F \cdot F}} \]

   7. Applied egg-rr 99.7%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.0001:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 0.0001)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (if (<= (* PI l_m) 20000000000000.0)
      (+ l_m (* (tan (* PI l_m)) (/ -1.0 (* 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) <= 0.0001) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else if ((((double) M_PI) * l_m) <= 20000000000000.0) {
		tmp = l_m + (tan((((double) M_PI) * l_m)) * (-1.0 / (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) <= 0.0001) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else if ((Math.PI * l_m) <= 20000000000000.0) {
		tmp = l_m + (Math.tan((Math.PI * l_m)) * (-1.0 / (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) <= 0.0001:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	elif (math.pi * l_m) <= 20000000000000.0:
		tmp = l_m + (math.tan((math.pi * l_m)) * (-1.0 / (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) <= 0.0001)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	elseif (Float64(pi * l_m) <= 20000000000000.0)
		tmp = Float64(l_m + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(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) <= 0.0001)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	elseif ((pi * l_m) <= 20000000000000.0)
		tmp = l_m + (tan((pi * l_m)) * (-1.0 / (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], 0.0001], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000000000.0], N[(l$95$m + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 83.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

   if 1.00000000000000005e-4 < (*.f64 (PI.f64) l) < 2e13

   1. Initial program 100.0%

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

      1. add-exp-log N/A

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

      2. add-sqr-sqrt N/A

         \[\leadsto e^{\log \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      3. log-prod N/A

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) + \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      4. flip-+ N/A

         \[\leadsto e^{\color{blue}{\frac{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. +-inverses N/A

         \[\leadsto e^{\frac{\color{blue}{0}}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      6. metadata-eval N/A

         \[\leadsto e^{\frac{\color{blue}{0 \cdot 0}}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      7. +-inverses N/A

         \[\leadsto e^{\frac{0 \cdot 0}{\color{blue}{0}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      8. associate-*r/ N/A

         \[\leadsto e^{\color{blue}{0 \cdot \frac{0}{0}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      9. +-inverses N/A

         \[\leadsto e^{0 \cdot \frac{\color{blue}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}}{0}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      10. +-inverses N/A

          \[\leadsto e^{0 \cdot \frac{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right) \cdot \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\color{blue}{\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      11. flip-+ N/A

          \[\leadsto e^{0 \cdot \color{blue}{\left(\log \left(\sqrt{\mathsf{PI}\left(\right)}\right) + \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      12. log-prod N/A

          \[\leadsto e^{0 \cdot \color{blue}{\log \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      13. add-sqr-sqrt N/A

          \[\leadsto e^{0 \cdot \log \color{blue}{\mathsf{PI}\left(\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      14. mul0-lft N/A

          \[\leadsto e^{\color{blue}{0}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      15. 1-exp N/A

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

   4. Applied egg-rr 100.0%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.0001:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (- (* 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) <= 20000000000000.0) {
		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) <= 20000000000000.0) {
		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) <= 20000000000000.0:
		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) <= 20000000000000.0)
		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) <= 20000000000000.0)
		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], 20000000000000.0], 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) < 2e13

   1. Initial program 83.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot {F}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot {F}^{2}}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot {F}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot {F}^{2}} \]

      9. lower-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot {F}^{2}} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(F \cdot F\right)}} \]

      11. lower-*.f64 83.8

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

   5. Simplified 83.8%

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

      1. lift-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \left(F \cdot F\right)} \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(F \cdot F\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(F \cdot F\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(F \cdot F\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(F \cdot F\right)}} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F \cdot F}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\color{blue}{F \cdot F}} \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F}}{F}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{F}}{F}} \]

   7. Applied egg-rr 87.3%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (- (* PI l_m) (/ (* 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) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((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) <= 20000000000000.0) {
		tmp = (Math.PI * l_m) - ((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) <= 20000000000000.0:
		tmp = (math.pi * l_m) - ((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) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(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) <= 20000000000000.0)
		tmp = (pi * l_m) - ((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], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $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) < 2e13

   1. Initial program 83.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Simplified 76.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (- (* 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) <= 20000000000000.0) {
		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) <= 20000000000000.0) {
		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) <= 20000000000000.0:
		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) <= 20000000000000.0)
		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) <= 20000000000000.0)
		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], 20000000000000.0], 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) < 2e13

   1. Initial program 83.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Simplified 76.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (* l_m (- PI (/ PI (* 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) <= 20000000000000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (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) <= 20000000000000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (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) <= 20000000000000.0:
		tmp = l_m * (math.pi - (math.pi / (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) <= 20000000000000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(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) <= 20000000000000.0)
		tmp = l_m * (pi - (pi / (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], 20000000000000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 83.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   8. Simplified 76.7%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.8% accurate, 22.5× 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 #s(literal 1 binary64) 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 79.3%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing

3. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 71.4

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

5. Simplified 71.4%

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

6. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   2. lower-PI.f64 74.0

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

8. Simplified 74.0%

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

9. Final simplification 74.0%

   \[\leadsto \pi \cdot \ell \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))