VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.3%

Time: 5.4s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

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: 76.4% accurate, 1.0× speedup

Math

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

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, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 1.05e+15)
     (- (* PI (fabs l)) (/ (/ (tan t_0) F) F))
     t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 1.05e+15) {
		tmp = (((double) M_PI) * fabs(l)) - ((tan(t_0) / F) / F);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.abs(l) * Math.PI;
	double tmp;
	if (Math.abs(l) <= 1.05e+15) {
		tmp = (Math.PI * Math.abs(l)) - ((Math.tan(t_0) / F) / F);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.fabs(l) * math.pi
	tmp = 0
	if math.fabs(l) <= 1.05e+15:
		tmp = (math.pi * math.fabs(l)) - ((math.tan(t_0) / F) / F)
	else:
		tmp = t_0
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 1.05e+15)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(tan(t_0) / F) / F));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = abs(l) * pi;
	tmp = 0.0;
	if (abs(l) <= 1.05e+15)
		tmp = (pi * abs(l)) - ((tan(t_0) / F) / F);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 1.05e+15], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Tan[t$95$0], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05e15

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

   if 1.05e15 < l

   1. Initial program 76.4%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|\ell\right|\\ t_1 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-22}:\\ \;\;\;\;t\_0 - \frac{\frac{t\_1}{F}}{F}\\ \mathbf{elif}\;\left|\ell\right| \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{\tan t\_1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* PI (fabs l))) (t_1 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 2e-22)
     (- t_0 (/ (/ t_1 F) F))
     (if (<= (fabs l) 1.05e+15) (- t_0 (/ (tan t_1) (* F F))) t_1)))))

C

double code(double F, double l) {
	double t_0 = ((double) M_PI) * fabs(l);
	double t_1 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 2e-22) {
		tmp = t_0 - ((t_1 / F) / F);
	} else if (fabs(l) <= 1.05e+15) {
		tmp = t_0 - (tan(t_1) / (F * F));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.PI * Math.abs(l);
	double t_1 = Math.abs(l) * Math.PI;
	double tmp;
	if (Math.abs(l) <= 2e-22) {
		tmp = t_0 - ((t_1 / F) / F);
	} else if (Math.abs(l) <= 1.05e+15) {
		tmp = t_0 - (Math.tan(t_1) / (F * F));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.pi * math.fabs(l)
	t_1 = math.fabs(l) * math.pi
	tmp = 0
	if math.fabs(l) <= 2e-22:
		tmp = t_0 - ((t_1 / F) / F)
	elif math.fabs(l) <= 1.05e+15:
		tmp = t_0 - (math.tan(t_1) / (F * F))
	else:
		tmp = t_1
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(pi * abs(l))
	t_1 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 2e-22)
		tmp = Float64(t_0 - Float64(Float64(t_1 / F) / F));
	elseif (abs(l) <= 1.05e+15)
		tmp = Float64(t_0 - Float64(tan(t_1) / Float64(F * F)));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = pi * abs(l);
	t_1 = abs(l) * pi;
	tmp = 0.0;
	if (abs(l) <= 2e-22)
		tmp = t_0 - ((t_1 / F) / F);
	elseif (abs(l) <= 1.05e+15)
		tmp = t_0 - (tan(t_1) / (F * F));
	else
		tmp = t_1;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 2e-22], N[(t$95$0 - N[(N[(t$95$1 / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.05e+15], N[(t$95$0 - N[(N[Tan[t$95$1], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-22

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 75.1%

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

   6. Applied rewrites 75.1%

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

   if 2.0000000000000001e-22 < l < 1.05e15

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.2%

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

   5. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. mult-flip N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. associate-/l/ N/A

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

      15. pow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-/.f64 76.7%

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 76.7%

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

   7. Applied rewrites 76.7%

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

   if 1.05e15 < l

   1. Initial program 76.4%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 3: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 3.5e+16)
     (- (* PI (fabs l)) (/ (/ t_0 F) F))
     t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 3.5e+16) {
		tmp = (((double) M_PI) * fabs(l)) - ((t_0 / F) / F);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.abs(l) * Math.PI;
	double tmp;
	if (Math.abs(l) <= 3.5e+16) {
		tmp = (Math.PI * Math.abs(l)) - ((t_0 / F) / F);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.fabs(l) * math.pi
	tmp = 0
	if math.fabs(l) <= 3.5e+16:
		tmp = (math.pi * math.fabs(l)) - ((t_0 / F) / F)
	else:
		tmp = t_0
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 3.5e+16)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(t_0 / F) / F));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = abs(l) * pi;
	tmp = 0.0;
	if (abs(l) <= 3.5e+16)
		tmp = (pi * abs(l)) - ((t_0 / F) / F);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 3.5e+16], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e16

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 75.1%

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

   6. Applied rewrites 75.1%

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

   if 3.5e16 < l

   1. Initial program 76.4%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 4: 92.8% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\left|\ell\right| \cdot \left(\pi - \frac{1}{F \cdot F} \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (*
 (copysign 1.0 l)
 (if (<= (fabs l) 3.5e+16)
   (* (fabs l) (- PI (* (/ 1.0 (* F F)) PI)))
   (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 3.5e+16) {
		tmp = fabs(l) * (((double) M_PI) - ((1.0 / (F * F)) * ((double) M_PI)));
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (Math.abs(l) <= 3.5e+16) {
		tmp = Math.abs(l) * (Math.PI - ((1.0 / (F * F)) * Math.PI));
	} else {
		tmp = Math.abs(l) * Math.PI;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	tmp = 0
	if math.fabs(l) <= 3.5e+16:
		tmp = math.fabs(l) * (math.pi - ((1.0 / (F * F)) * math.pi))
	else:
		tmp = math.fabs(l) * math.pi
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	tmp = 0.0
	if (abs(l) <= 3.5e+16)
		tmp = Float64(abs(l) * Float64(pi - Float64(Float64(1.0 / Float64(F * F)) * pi)));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (abs(l) <= 3.5e+16)
		tmp = abs(l) * (pi - ((1.0 / (F * F)) * pi));
	else
		tmp = abs(l) * pi;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 3.5e+16], N[(N[Abs[l], $MachinePrecision] * N[(Pi - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e16

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-pow.f64 69.4%

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

   6. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 69.4%

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

   8. Applied rewrites 69.4%

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

   if 3.5e16 < l

   1. Initial program 76.4%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 5: 92.8% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot \left|\ell\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (*
 (copysign 1.0 l)
 (if (<= (fabs l) 3.5e+16)
   (* (- PI (/ PI (* F F))) (fabs l))
   (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 3.5e+16) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * fabs(l);
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (Math.abs(l) <= 3.5e+16) {
		tmp = (Math.PI - (Math.PI / (F * F))) * Math.abs(l);
	} else {
		tmp = Math.abs(l) * Math.PI;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	tmp = 0
	if math.fabs(l) <= 3.5e+16:
		tmp = (math.pi - (math.pi / (F * F))) * math.fabs(l)
	else:
		tmp = math.fabs(l) * math.pi
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	tmp = 0.0
	if (abs(l) <= 3.5e+16)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * abs(l));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (abs(l) <= 3.5e+16)
		tmp = (pi - (pi / (F * F))) * abs(l);
	else
		tmp = abs(l) * pi;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 3.5e+16], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e16

   1. Initial program 76.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.2%

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-pow.f64 69.4%

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.4%

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 69.4%

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

   8. Applied rewrites 69.4%

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

   if 3.5e16 < l

   1. Initial program 76.4%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 6: 74.0% accurate, 22.5× speedup

Math

\[\ell \cdot \pi \]

FPCore

(FPCore (F l)
  :precision binary64
  (* l PI))

C

double code(double F, double l) {
	return l * ((double) M_PI);
}

Java

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

Python

def code(F, l):
	return l * math.pi

Julia

function code(F, l)
	return Float64(l * pi)
end

MATLAB

function tmp = code(F, l)
	tmp = l * pi;
end

Wolfram

code[F_, l_] := N[(l * Pi), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in F around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 74.0%

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

4. Applied rewrites 74.0%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))