VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 99.3%

Time: 3.4s

Alternatives: 9

Speedup: 2.3×

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: 76.1% 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, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{1}{\frac{F}{\frac{\tan t\_0}{F}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 1.85e+15)
     (- (* PI (fabs l)) (/ 1.0 (/ F (/ (tan t_0) F))))
     t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 1.85e+15) {
		tmp = (((double) M_PI) * fabs(l)) - (1.0 / (F / (tan(t_0) / 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) <= 1.85e+15) {
		tmp = (Math.PI * Math.abs(l)) - (1.0 / (F / (Math.tan(t_0) / 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) <= 1.85e+15:
		tmp = (math.pi * math.fabs(l)) - (1.0 / (F / (math.tan(t_0) / 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) <= 1.85e+15)
		tmp = Float64(Float64(pi * abs(l)) - Float64(1.0 / Float64(F / Float64(tan(t_0) / 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) <= 1.85e+15)
		tmp = (pi * abs(l)) - (1.0 / (F / (tan(t_0) / 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], 1.85e+15], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(F / N[(N[Tan[t$95$0], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.85e15

   1. Initial program 76.1%

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

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-/.f64 81.8%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.8%

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

   3. Applied rewrites 81.8%

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

   if 1.85e15 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan t\_0}{F}, \frac{-1}{F}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 1.85e+15)
     (fma (/ (tan t_0) F) (/ -1.0 F) t_0)
     t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 1.85e+15) {
		tmp = fma((tan(t_0) / F), (-1.0 / F), t_0);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 1.85e+15)
		tmp = fma(Float64(tan(t_0) / F), Float64(-1.0 / F), t_0);
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * 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], 1.85e+15], N[(N[(N[Tan[t$95$0], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.85e15

   1. Initial program 76.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 81.8%

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

   if 1.85e15 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 1.85e+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) <= 1.85e+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) <= 1.85e+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) <= 1.85e+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) <= 1.85e+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) <= 1.85e+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], 1.85e+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 < 1.85e15

   1. Initial program 76.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.8%

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

      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 81.8%

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

   3. Applied rewrites 81.8%

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

   if 1.85e15 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 4: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 4500000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{F}, \frac{t\_0}{F}, \pi \cdot \left|\ell\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 4500000000000.0)
     (fma (/ -1.0 F) (/ t_0 F) (* PI (fabs l)))
     t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 4500000000000.0) {
		tmp = fma((-1.0 / F), (t_0 / F), (((double) M_PI) * fabs(l)));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 4500000000000.0)
		tmp = fma(Float64(-1.0 / F), Float64(t_0 / F), Float64(pi * abs(l)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * 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], 4500000000000.0], N[(N[(-1.0 / F), $MachinePrecision] * N[(t$95$0 / F), $MachinePrecision] + N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.5e12

   1. Initial program 76.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.8%

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

      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 81.8%

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

   3. Applied rewrites 81.8%

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.1%

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

   8. Applied rewrites 74.1%

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

   if 4.5e12 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 4500000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1.0 l)
   (if (<= (fabs l) 4500000000000.0)
     (- (* 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) <= 4500000000000.0) {
		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) <= 4500000000000.0) {
		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) <= 4500000000000.0:
		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) <= 4500000000000.0)
		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) <= 4500000000000.0)
		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], 4500000000000.0], 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 < 4.5e12

   1. Initial program 76.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.8%

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

      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 81.8%

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

   3. Applied rewrites 81.8%

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

   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 74.1%

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

   6. Applied rewrites 74.1%

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

   if 4.5e12 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 6: 92.6% accurate, 2.3× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 4500000000000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot \left|\ell\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (*
 (copysign 1.0 l)
 (if (<= (fabs l) 4500000000000.0)
   (* (- PI (/ PI (* F F))) (fabs l))
   (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 4500000000000.0) {
		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) <= 4500000000000.0) {
		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) <= 4500000000000.0:
		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) <= 4500000000000.0)
		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) <= 4500000000000.0)
		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], 4500000000000.0], 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 < 4.5e12

   1. Initial program 76.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   3. 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 68.5%

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. 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 68.5%

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

      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. lift-*.f64 68.5%

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

   6. Applied rewrites 68.5%

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

   if 4.5e12 < l

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 7: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\pi}{F} \cdot \frac{\left|\ell\right|}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* PI (fabs l))))
  (*
   (copysign 1.0 l)
   (if (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -2e-226)
     (* (/ (- PI) F) (/ (fabs l) F))
     (* (fabs l) PI)))))

C

double code(double F, double l) {
	double t_0 = ((double) M_PI) * fabs(l);
	double tmp;
	if ((t_0 - ((1.0 / (F * F)) * tan(t_0))) <= -2e-226) {
		tmp = (-((double) M_PI) / F) * (fabs(l) / F);
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.PI * Math.abs(l);
	double tmp;
	if ((t_0 - ((1.0 / (F * F)) * Math.tan(t_0))) <= -2e-226) {
		tmp = (-Math.PI / F) * (Math.abs(l) / F);
	} else {
		tmp = Math.abs(l) * Math.PI;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.pi * math.fabs(l)
	tmp = 0
	if (t_0 - ((1.0 / (F * F)) * math.tan(t_0))) <= -2e-226:
		tmp = (-math.pi / F) * (math.fabs(l) / F)
	else:
		tmp = math.fabs(l) * math.pi
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(pi * abs(l))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(1.0 / Float64(F * F)) * tan(t_0))) <= -2e-226)
		tmp = Float64(Float64(Float64(-pi) / F) * Float64(abs(l) / F));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = pi * abs(l);
	tmp = 0.0;
	if ((t_0 - ((1.0 / (F * F)) * tan(t_0))) <= -2e-226)
		tmp = (-pi / F) * (abs(l) / F);
	else
		tmp = abs(l) * pi;
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$0 - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-226], N[(N[((-Pi) / F), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.9999999999999998e-226

   1. Initial program 76.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   3. 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 68.5%

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. 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. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-to-fraction N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 38.2%

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

   6. Applied rewrites 38.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 21.1%

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

   9. Applied rewrites 21.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-neg.f64 N/A

           \[\leadsto \frac{-\pi}{F} \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\ell}{F}\right)\right) \]

   11. Applied rewrites 26.4%

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

   if -1.9999999999999998e-226 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 8: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-3.141592653589793 \cdot \left|\ell\right|}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* PI (fabs l))))
  (*
   (copysign 1.0 l)
   (if (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -2e-226)
     (/ (* -3.141592653589793 (fabs l)) (* F F))
     (* (fabs l) PI)))))

C

double code(double F, double l) {
	double t_0 = ((double) M_PI) * fabs(l);
	double tmp;
	if ((t_0 - ((1.0 / (F * F)) * tan(t_0))) <= -2e-226) {
		tmp = (-3.141592653589793 * fabs(l)) / (F * F);
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.PI * Math.abs(l);
	double tmp;
	if ((t_0 - ((1.0 / (F * F)) * Math.tan(t_0))) <= -2e-226) {
		tmp = (-3.141592653589793 * Math.abs(l)) / (F * F);
	} else {
		tmp = Math.abs(l) * Math.PI;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.pi * math.fabs(l)
	tmp = 0
	if (t_0 - ((1.0 / (F * F)) * math.tan(t_0))) <= -2e-226:
		tmp = (-3.141592653589793 * math.fabs(l)) / (F * F)
	else:
		tmp = math.fabs(l) * math.pi
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(pi * abs(l))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(1.0 / Float64(F * F)) * tan(t_0))) <= -2e-226)
		tmp = Float64(Float64(-3.141592653589793 * abs(l)) / Float64(F * F));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = pi * abs(l);
	tmp = 0.0;
	if ((t_0 - ((1.0 / (F * F)) * tan(t_0))) <= -2e-226)
		tmp = (-3.141592653589793 * abs(l)) / (F * F);
	else
		tmp = abs(l) * pi;
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$0 - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-226], N[(N[(-3.141592653589793 * N[Abs[l], $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.9999999999999998e-226

   1. Initial program 76.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   3. 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 68.5%

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. 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. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-to-fraction N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 38.2%

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

   6. Applied rewrites 38.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 21.1%

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

   9. Applied rewrites 21.1%

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

   10. Evaluated real constant 21.1%

       \[\leadsto \frac{-3.141592653589793 \cdot \ell}{F \cdot F} \]

   if -1.9999999999999998e-226 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.1%

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

   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 73.6%

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

   4. Applied rewrites 73.6%

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

Alternative 9: 73.6% accurate, 14.2× 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 76.1%

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

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 73.6%

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

4. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025213 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))