VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.0% → 99.2%

Time: 6.1s

Alternatives: 9

Speedup: 2.7×

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: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot l\_m\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 3.3e+15)
    (- (* PI l_m) (* (/ 1.0 F) (/ (tan (* PI l_m)) F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.3e+15) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / F) * (tan((((double) M_PI) * l_m)) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.3e+15) {
		tmp = (Math.PI * l_m) - ((1.0 / F) * (Math.tan((Math.PI * l_m)) / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 3.3e+15:
		tmp = (math.pi * l_m) - ((1.0 / F) * (math.tan((math.pi * l_m)) / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 3.3e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / F) * Float64(tan(Float64(pi * l_m)) / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 3.3e+15)
		tmp = (pi * l_m) - ((1.0 / F) * (tan((pi * l_m)) / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 3.3e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] * N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.3e15

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-tan.f64 98.9

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

   3. Applied rewrites 98.9%

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

   if 3.3e15 < l

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 3.3e+15)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.3e+15) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.3e+15) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 3.3e+15:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 3.3e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 3.3e+15)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 3.3e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.3e15

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative N/A

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

      13. quot-tan N/A

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

      14. quot-tan N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 98.9%

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

   if 3.3e15 < l

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.15 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\mathsf{fma}\left(0.3333333333333333 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), l\_m \cdot l\_m, \pi\right) \cdot l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 3.15e+15)
    (-
     (* PI l_m)
     (/
      (/
       (* (fma (* 0.3333333333333333 (* (* PI PI) PI)) (* l_m 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 (l_m <= 3.15e+15) {
		tmp = (((double) M_PI) * l_m) - (((fma((0.3333333333333333 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (l_m * 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 (l_m <= 3.15e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(fma(Float64(0.3333333333333333 * Float64(Float64(pi * pi) * pi)), Float64(l_m * l_m), pi) * l_m) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 3.15e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(N[(N[(0.3333333333333333 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision] + Pi), $MachinePrecision] * 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 l < 3.15e15

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative N/A

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

      13. quot-tan N/A

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

      14. quot-tan N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}}{F}}{F} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \pi \cdot \ell - \frac{\frac{\left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{\ell}}{F}}{F} \]

      2. lower-*.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\frac{\left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{\ell}}{F}}{F} \]

   6. Applied rewrites 97.2%

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

   if 3.15e15 < l

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 4: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.15 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 3.15e+15) (- (* 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 (l_m <= 3.15e+15) {
		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 (l_m <= 3.15e+15) {
		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 l_m <= 3.15e+15:
		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 (l_m <= 3.15e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 3.15e+15)
		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[l$95$m, 3.15e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.15e15

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative N/A

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

      13. quot-tan N/A

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

      14. quot-tan N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-PI.f64 97.2

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

   6. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-/.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if 3.15e15 < l

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 5: 89.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\pi \cdot \frac{\frac{-l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -5e-284)
    (* 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -5e-284:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -5e-284)
		tmp = Float64(pi * Float64(Float64(Float64(-l_m) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (((pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -5e-284)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-284], N[(Pi * N[(N[((-l$95$m) / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -4.99999999999999973e-284

   1. Initial program 57.5%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

   9. Applied rewrites 55.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l/ N/A

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

       6. associate-*r/ N/A

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

       7. mul-1-neg N/A

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

       8. distribute-neg-frac N/A

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

       9. pow2 N/A

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

       10. associate-/r* N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-neg.f64 71.6

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

   11. Applied rewrites 71.6%

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

   if -4.99999999999999973e-284 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 85.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 6: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{\left(-l\_m\right) \cdot \pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -5e-284)
    (/ (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -5e-284:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -5e-284)
		tmp = Float64(Float64(Float64(-l_m) * pi) / Float64(F * F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (((pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -5e-284)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-284], N[(N[((-l$95$m) * Pi), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -4.99999999999999973e-284

   1. Initial program 57.5%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if -4.99999999999999973e-284 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 85.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 7: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\left(-\pi\right) \cdot \frac{l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -5e-284)
    (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -5e-284:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -5e-284)
		tmp = Float64(Float64(-pi) * Float64(l_m / Float64(F * F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (((pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -5e-284)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-284], 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)))) < -4.99999999999999973e-284

   1. Initial program 57.5%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. lift-neg.f64 N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 56.9

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

   9. Applied rewrites 56.9%

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

   if -4.99999999999999973e-284 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 85.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 8: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -5e-284)
    (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -5e-284) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -5e-284:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -5e-284)
		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) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -5e-284)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-284], 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)))) < -4.99999999999999973e-284

   1. Initial program 57.5%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

   if -4.99999999999999973e-284 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 85.6%

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

   2. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 9: 73.8% accurate, 13.6× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* PI l_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 73.8

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

4. Applied rewrites 73.8%

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

Reproduce

herbie shell --seed 2025134 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))