VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.3%

Time: 16.2s

Alternatives: 13

Speedup: 3.7×

• could not determine a ground truth

  1. F = 7.3579658613052665e-292
  2. l = -3727329285076699.5

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: 76.9% 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 2e+14)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e+14) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 2e+14) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = (l_m * Math.sqrt(Math.sqrt(Math.PI))) * Math.sqrt((Math.PI * Math.sqrt(Math.PI)));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 2e+14:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = (l_m * math.sqrt(math.sqrt(math.pi))) * math.sqrt((math.pi * math.sqrt(math.pi)))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e+14)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e+14)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = (l_m * sqrt(sqrt(pi))) * sqrt((pi * sqrt(pi)));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e+14], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 78.0%

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

      1. associate-/r* N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 63.7%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\pi \cdot \frac{l\_m}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -5e-273)
    (* 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)))) <= -5e-273) {
		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)))) <= -5e-273) {
		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)))) <= -5e-273:
		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)))) <= -5e-273)
		tmp = Float64(pi * Float64(l_m / Float64(F * 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 (((pi * l_m) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -5e-273)
		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], -5e-273], N[(Pi * N[(l$95$m / 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)))) < -4.99999999999999965e-273

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 63.2

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

   5. Applied rewrites 63.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 22.6

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

   8. Applied rewrites 22.6%

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

      1. lift-PI.f64 N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 21.9

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

   10. Applied rewrites 21.9%

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

       1. lift-neg.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. clear-num N/A

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

       5. un-div-inv N/A

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

       6. frac-2neg N/A

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

       7. associate-/r/ N/A

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

       8. lift-*.f64 N/A

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

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

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

       10. lift-neg.f64 N/A

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

       11. associate-/l/ N/A

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

       12. lift-neg.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. frac-2neg N/A

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

       15. lower-*.f64 N/A

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

       16. associate-/l/ N/A

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

       17. lift-*.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower-neg.f64 22.7

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

   12. Applied rewrites 22.7%

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

   if -4.99999999999999965e-273 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.6%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\pi \cdot \frac{\ell}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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)))) -5e-273)
    (* (- 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)))) <= -5e-273) {
		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)))) <= -5e-273) {
		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)))) <= -5e-273:
		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)))) <= -5e-273)
		tmp = Float64(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 (((pi * l_m) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -5e-273)
		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], -5e-273], 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)))) < -4.99999999999999965e-273

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 63.2

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

   5. Applied rewrites 63.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 22.6

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

   8. Applied rewrites 22.6%

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

      1. lift-PI.f64 N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 21.9

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

   10. Applied rewrites 21.9%

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

   if -4.99999999999999965e-273 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.6%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 2.0× speedup

Math

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

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-8)
    (- (* PI l_m) (/ (/ (* PI l_m) F) F))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = (((double) M_PI) * l_m) - (((((double) M_PI) * l_m) / F) / F);
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e-8) {
		tmp = (Math.PI * l_m) - (((Math.PI * l_m) / F) / F);
	} else {
		tmp = (l_m * Math.sqrt(Math.sqrt(Math.PI))) * Math.sqrt((Math.PI * Math.sqrt(Math.PI)));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e-8:
		tmp = (math.pi * l_m) - (((math.pi * l_m) / F) / F)
	else:
		tmp = (l_m * math.sqrt(math.sqrt(math.pi))) * math.sqrt((math.pi * math.sqrt(math.pi)))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi * l_m) / F) / F));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e-8)
		tmp = (pi * l_m) - (((pi * l_m) / F) / F);
	else
		tmp = (l_m * sqrt(sqrt(pi))) * sqrt((pi * sqrt(pi)));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.6%

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

Alternative 5: 91.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-272}:\\ \;\;\;\;-\frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \pi \cdot \frac{l\_m}{F \cdot \left(-F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 1e-272)
    (- (/ (/ (* PI l_m) F) F))
    (if (<= (* PI l_m) 1e-8)
      (fma 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) <= 1e-272) {
		tmp = -(((((double) M_PI) * l_m) / F) / F);
	} else if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = fma(((double) M_PI), l_m, (((double) M_PI) * (l_m / (F * -F))));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-272)
		tmp = Float64(-Float64(Float64(Float64(pi * l_m) / F) / F));
	elseif (Float64(pi * l_m) <= 1e-8)
		tmp = fma(pi, l_m, Float64(pi * Float64(l_m / Float64(F * Float64(-F)))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-272], (-N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * l$95$m + N[(Pi * N[(l$95$m / 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) < 9.9999999999999993e-273

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 62.2

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

   5. Applied rewrites 62.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 21.3

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

   8. Applied rewrites 21.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-/.f64 32.5

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 32.6

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

   10. Applied rewrites 32.6%

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

   if 9.9999999999999993e-273 < (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 90.9

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-272}:\\ \;\;\;\;-\frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \pi \cdot \frac{\ell}{F \cdot \left(-F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-272}:\\ \;\;\;\;-\frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 1e-272)
    (- (/ (/ (* PI l_m) F) F))
    (if (<= (* PI l_m) 1e-8) (* PI (- l_m (/ 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) <= 1e-272) {
		tmp = -(((((double) M_PI) * l_m) / F) / F);
	} else if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = ((double) M_PI) * (l_m - (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) <= 1e-272) {
		tmp = -(((Math.PI * l_m) / F) / F);
	} else if ((Math.PI * l_m) <= 1e-8) {
		tmp = Math.PI * (l_m - (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) <= 1e-272:
		tmp = -(((math.pi * l_m) / F) / F)
	elif (math.pi * l_m) <= 1e-8:
		tmp = math.pi * (l_m - (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) <= 1e-272)
		tmp = Float64(-Float64(Float64(Float64(pi * l_m) / F) / F));
	elseif (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(pi * Float64(l_m - 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) <= 1e-272)
		tmp = -(((pi * l_m) / F) / F);
	elseif ((pi * l_m) <= 1e-8)
		tmp = pi * (l_m - (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], 1e-272], (-N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * N[(l$95$m - N[(l$95$m / 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) < 9.9999999999999993e-273

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 62.2

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

   5. Applied rewrites 62.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 21.3

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

   8. Applied rewrites 21.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-/.f64 32.5

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 32.6

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

   10. Applied rewrites 32.6%

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

   if 9.9999999999999993e-273 < (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-272}:\\ \;\;\;\;-\frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-272}:\\ \;\;\;\;\frac{\pi}{F \cdot \frac{F}{-l\_m}}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 1e-272)
    (/ PI (* F (/ F (- l_m))))
    (if (<= (* PI l_m) 1e-8) (* PI (- l_m (/ 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) <= 1e-272) {
		tmp = ((double) M_PI) / (F * (F / -l_m));
	} else if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = ((double) M_PI) * (l_m - (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) <= 1e-272) {
		tmp = Math.PI / (F * (F / -l_m));
	} else if ((Math.PI * l_m) <= 1e-8) {
		tmp = Math.PI * (l_m - (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) <= 1e-272:
		tmp = math.pi / (F * (F / -l_m))
	elif (math.pi * l_m) <= 1e-8:
		tmp = math.pi * (l_m - (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) <= 1e-272)
		tmp = Float64(pi / Float64(F * Float64(F / Float64(-l_m))));
	elseif (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(pi * Float64(l_m - 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) <= 1e-272)
		tmp = pi / (F * (F / -l_m));
	elseif ((pi * l_m) <= 1e-8)
		tmp = pi * (l_m - (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], 1e-272], N[(Pi / N[(F * N[(F / (-l$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * N[(l$95$m - N[(l$95$m / 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) < 9.9999999999999993e-273

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 62.2

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

   5. Applied rewrites 62.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 21.3

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

   8. Applied rewrites 21.3%

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

      1. lift-PI.f64 N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 20.7

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

   10. Applied rewrites 20.7%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-lft-neg-out N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. distribute-frac-neg2 N/A

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

       10. lift-*.f64 N/A

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

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

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

       12. lift-neg.f64 N/A

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

       13. associate-/l/ N/A

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

       14. lift-/.f64 N/A

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

       15. clear-num N/A

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

       16. lift-neg.f64 N/A

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

       17. neg-mul-1 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. associate-/l* N/A

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

       21. times-frac N/A

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

   12. Applied rewrites 32.5%

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

   if 9.9999999999999993e-273 < (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-272}:\\ \;\;\;\;\frac{\pi}{F \cdot \frac{F}{-\ell}}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-272}:\\ \;\;\;\;\left(-\frac{l\_m}{F}\right) \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 1e-272)
    (* (- (/ l_m F)) (/ PI F))
    (if (<= (* PI l_m) 1e-8) (* PI (- l_m (/ 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) <= 1e-272) {
		tmp = -(l_m / F) * (((double) M_PI) / F);
	} else if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = ((double) M_PI) * (l_m - (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) <= 1e-272) {
		tmp = -(l_m / F) * (Math.PI / F);
	} else if ((Math.PI * l_m) <= 1e-8) {
		tmp = Math.PI * (l_m - (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) <= 1e-272:
		tmp = -(l_m / F) * (math.pi / F)
	elif (math.pi * l_m) <= 1e-8:
		tmp = math.pi * (l_m - (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) <= 1e-272)
		tmp = Float64(Float64(-Float64(l_m / F)) * Float64(pi / F));
	elseif (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(pi * Float64(l_m - 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) <= 1e-272)
		tmp = -(l_m / F) * (pi / F);
	elseif ((pi * l_m) <= 1e-8)
		tmp = pi * (l_m - (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], 1e-272], N[((-N[(l$95$m / F), $MachinePrecision]) * N[(Pi / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * N[(l$95$m - N[(l$95$m / 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) < 9.9999999999999993e-273

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 62.2

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

   5. Applied rewrites 62.2%

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

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 21.3

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

   8. Applied rewrites 21.3%

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

      1. lift-PI.f64 N/A

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

      2. distribute-lft-neg-in N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 32.5

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

   10. Applied rewrites 32.5%

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

   if 9.9999999999999993e-273 < (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 90.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 90.9

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

   5. Applied rewrites 90.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-272}:\\ \;\;\;\;\left(-\frac{\ell}{F}\right) \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 2.9× speedup

Math

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

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) 1e-8) (- (* 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) <= 1e-8) {
		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) <= 1e-8) {
		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) <= 1e-8:
		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) <= 1e-8)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(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) <= 1e-8)
		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], 1e-8], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $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) < 1e-8

   1. Initial program 77.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 86.4%

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

Alternative 10: 97.9% accurate, 2.9× speedup

Math

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

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 (<= (* PI l_m) 1e-8) (- (* PI l_m) (/ 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 ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = (((double) M_PI) * l_m) - (((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 ((Math.PI * l_m) <= 1e-8) {
		tmp = (Math.PI * l_m) - (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 (math.pi * l_m) <= 1e-8:
		tmp = (math.pi * l_m) - (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 (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(Float64(pi * l_m) - Float64(pi / Float64(F * Float64(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 ((pi * l_m) <= 1e-8)
		tmp = (pi * l_m) - (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[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(Pi / N[(F * N[(F / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. *-lft-identity N/A

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

      20. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.5

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

   9. Applied rewrites 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 86.4%

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

Alternative 11: 92.7% accurate, 3.7× speedup

Math

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

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) 1e-8) (* PI (- l_m (/ 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) <= 1e-8) {
		tmp = ((double) M_PI) * (l_m - (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) <= 1e-8) {
		tmp = Math.PI * (l_m - (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) <= 1e-8:
		tmp = math.pi * (l_m - (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) <= 1e-8)
		tmp = Float64(pi * Float64(l_m - 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) <= 1e-8)
		tmp = pi * (l_m - (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], 1e-8], N[(Pi * N[(l$95$m - N[(l$95$m / 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) < 1e-8

   1. Initial program 77.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.2% accurate, 3.7× speedup

Math

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

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) 1e-8) (* 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) <= 1e-8) {
		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) <= 1e-8) {
		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) <= 1e-8:
		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) <= 1e-8)
		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) <= 1e-8)
		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], 1e-8], 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) < 1e-8

   1. Initial program 77.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   4. 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 71.7

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

   5. Applied rewrites 71.7%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.1% 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 74.8%

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

3. Taylor expanded in l around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 71.7

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

5. Applied rewrites 71.7%

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

6. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))