VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 99.2%

Time: 3.8s

Alternatives: 8

Speedup: 4.4×

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: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{-F} \cdot \frac{\tan \left(\pi \cdot l\_m\right)}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 3e+14)
    (- (* PI l_m) (* (/ 1.0 (- F)) (/ (tan (* PI 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 (l_m <= 3e+14) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / -F) * (tan((((double) M_PI) * 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 (l_m <= 3e+14) {
		tmp = (Math.PI * l_m) - ((1.0 / -F) * (Math.tan((Math.PI * 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 l_m <= 3e+14:
		tmp = (math.pi * l_m) - ((1.0 / -F) * (math.tan((math.pi * 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 (l_m <= 3e+14)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(-F)) * Float64(tan(Float64(pi * l_m)) / Float64(-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 (l_m <= 3e+14)
		tmp = (pi * l_m) - ((1.0 / -F) * (tan((pi * 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[l$95$m, 3e+14], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / (-F)), $MachinePrecision] * N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3e14

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 99.0

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

   3. Applied rewrites 99.0%

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

   if 3e14 < l

   1. Initial program 63.9%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 2: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+191}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-262}:\\ \;\;\;\;\frac{-\pi}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m))))))
   (*
    l_s
    (if (<= t_0 -5e+191)
      (* PI l_m)
      (if (<= t_0 -2e-262) (* (/ (- PI) (* F F)) l_m) (* 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 t_0 = (((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)));
	double tmp;
	if (t_0 <= -5e+191) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -2e-262) {
		tmp = (-((double) M_PI) / (F * F)) * l_m;
	} 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 t_0 = (Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)));
	double tmp;
	if (t_0 <= -5e+191) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -2e-262) {
		tmp = (-Math.PI / (F * F)) * l_m;
	} 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):
	t_0 = (math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))
	tmp = 0
	if t_0 <= -5e+191:
		tmp = math.pi * l_m
	elif t_0 <= -2e-262:
		tmp = (-math.pi / (F * F)) * l_m
	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)
	t_0 = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m))))
	tmp = 0.0
	if (t_0 <= -5e+191)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -2e-262)
		tmp = Float64(Float64(Float64(-pi) / Float64(F * F)) * l_m);
	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)
	t_0 = (pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)));
	tmp = 0.0;
	if (t_0 <= -5e+191)
		tmp = pi * l_m;
	elseif (t_0 <= -2e-262)
		tmp = (-pi / (F * F)) * l_m;
	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_] := Block[{t$95$0 = N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[t$95$0, -5e+191], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -2e-262], N[(N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision] * l$95$m), $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)))) < -5.0000000000000002e191 or -2.00000000000000002e-262 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 73.9%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.5

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

   4. Applied rewrites 83.5%

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

   if -5.0000000000000002e191 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2.00000000000000002e-262

   1. Initial program 91.7%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in F around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-PI.f64 89.0

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

   7. Applied rewrites 89.0%

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

Alternative 3: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+191}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-262}:\\ \;\;\;\;\frac{-l\_m \cdot \pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m))))))
   (*
    l_s
    (if (<= t_0 -5e+191)
      (* PI l_m)
      (if (<= t_0 -2e-262) (/ (- (* 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 t_0 = (((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)));
	double tmp;
	if (t_0 <= -5e+191) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -2e-262) {
		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 t_0 = (Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)));
	double tmp;
	if (t_0 <= -5e+191) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -2e-262) {
		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):
	t_0 = (math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))
	tmp = 0
	if t_0 <= -5e+191:
		tmp = math.pi * l_m
	elif t_0 <= -2e-262:
		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)
	t_0 = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m))))
	tmp = 0.0
	if (t_0 <= -5e+191)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -2e-262)
		tmp = Float64(Float64(-Float64(l_m * 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)
	t_0 = (pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)));
	tmp = 0.0;
	if (t_0 <= -5e+191)
		tmp = pi * l_m;
	elseif (t_0 <= -2e-262)
		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_] := Block[{t$95$0 = N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[t$95$0, -5e+191], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -2e-262], N[((-N[(l$95$m * Pi), $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)))) < -5.0000000000000002e191 or -2.00000000000000002e-262 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 73.9%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.5

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

   4. Applied rewrites 83.5%

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

   if -5.0000000000000002e191 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2.00000000000000002e-262

   1. Initial program 91.7%

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

   2. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 89.0

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

   10. Applied rewrites 89.0%

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

Alternative 4: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 58000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{-F} \cdot \frac{\pi \cdot l\_m}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 58000000000000.0)
    (- (* PI l_m) (* (/ 1.0 (- F)) (/ (* PI 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 (l_m <= 58000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / -F) * ((((double) M_PI) * 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 (l_m <= 58000000000000.0) {
		tmp = (Math.PI * l_m) - ((1.0 / -F) * ((Math.PI * 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 l_m <= 58000000000000.0:
		tmp = (math.pi * l_m) - ((1.0 / -F) * ((math.pi * 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 (l_m <= 58000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(-F)) * Float64(Float64(pi * l_m) / Float64(-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 (l_m <= 58000000000000.0)
		tmp = (pi * l_m) - ((1.0 / -F) * ((pi * 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[l$95$m, 58000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / (-F)), $MachinePrecision] * N[(N[(Pi * l$95$m), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.8e13

   1. Initial program 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.5

         \[\leadsto \pi \cdot \ell - \frac{1}{-F} \cdot \frac{\pi \cdot \ell}{-F} \]

   6. Applied rewrites 97.5%

      \[\leadsto \pi \cdot \ell - \frac{1}{-F} \cdot \frac{\color{blue}{\pi \cdot \ell}}{-F} \]

   if 5.8e13 < l

   1. Initial program 64.0%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 5: 92.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{\frac{\left(F \cdot F\right) \cdot \pi - \pi}{F} \cdot l\_m}{F}\\ \mathbf{elif}\;l\_m \leq 58000000000000:\\ \;\;\;\;\pi \cdot l\_m - l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 5.4e-244)
    (/ (* (/ (- (* (* F F) PI) PI) F) l_m) F)
    (if (<= l_m 58000000000000.0)
      (- (* PI l_m) (* 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 (l_m <= 5.4e-244) {
		tmp = (((((F * F) * ((double) M_PI)) - ((double) M_PI)) / F) * l_m) / F;
	} else if (l_m <= 58000000000000.0) {
		tmp = (((double) M_PI) * l_m) - (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 (l_m <= 5.4e-244) {
		tmp = (((((F * F) * Math.PI) - Math.PI) / F) * l_m) / F;
	} else if (l_m <= 58000000000000.0) {
		tmp = (Math.PI * l_m) - (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 l_m <= 5.4e-244:
		tmp = (((((F * F) * math.pi) - math.pi) / F) * l_m) / F
	elif l_m <= 58000000000000.0:
		tmp = (math.pi * l_m) - (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 (l_m <= 5.4e-244)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(F * F) * pi) - pi) / F) * l_m) / F);
	elseif (l_m <= 58000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(l_m * 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 (l_m <= 5.4e-244)
		tmp = (((((F * F) * pi) - pi) / F) * l_m) / F;
	elseif (l_m <= 58000000000000.0)
		tmp = (pi * l_m) - (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[l$95$m, 5.4e-244], N[(N[(N[(N[(N[(N[(F * F), $MachinePrecision] * Pi), $MachinePrecision] - Pi), $MachinePrecision] / F), $MachinePrecision] * l$95$m), $MachinePrecision] / F), $MachinePrecision], If[LessEqual[l$95$m, 58000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(l$95$m * N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 5.3999999999999999e-244

   1. Initial program 80.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 59.4

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

   7. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 77.4%

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

   if 5.3999999999999999e-244 < l < 5.8e13

   1. Initial program 90.0%

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

   2. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

   if 5.8e13 < l

   1. Initial program 64.0%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 6: 92.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 58000000000000:\\ \;\;\;\;\pi \cdot l\_m - l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 58000000000000.0)
    (- (* PI l_m) (* 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 (l_m <= 58000000000000.0) {
		tmp = (((double) M_PI) * l_m) - (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 (l_m <= 58000000000000.0) {
		tmp = (Math.PI * l_m) - (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 l_m <= 58000000000000.0:
		tmp = (math.pi * l_m) - (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 (l_m <= 58000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(l_m * 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 (l_m <= 58000000000000.0)
		tmp = (pi * l_m) - (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[l$95$m, 58000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(l$95$m * 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 l < 5.8e13

   1. Initial program 88.0%

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

   2. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if 5.8e13 < l

   1. Initial program 64.0%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 7: 92.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 58000000000000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 58000000000000.0) (* (- PI (/ PI (* F F))) l_m) (* 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 (l_m <= 58000000000000.0) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * l_m;
	} 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 (l_m <= 58000000000000.0) {
		tmp = (Math.PI - (Math.PI / (F * F))) * l_m;
	} 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 l_m <= 58000000000000.0:
		tmp = (math.pi - (math.pi / (F * F))) * l_m
	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 (l_m <= 58000000000000.0)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * l_m);
	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 (l_m <= 58000000000000.0)
		tmp = (pi - (pi / (F * F))) * l_m;
	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[l$95$m, 58000000000000.0], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.8e13

   1. Initial program 88.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if 5.8e13 < l

   1. Initial program 64.0%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 8: 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 76.2%

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

2. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 73.8

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

4. Applied rewrites 73.8%

   \[\leadsto \color{blue}{\pi \cdot \ell} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025099 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))