VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 99.2%

Time: 2.1s

Alternatives: 8

Speedup: 0.9×

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 (* 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 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.9% 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 (* 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 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2900000000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{-1}{\frac{-F}{\tan t\_0} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1 l)
   (if (<= (fabs l) 2900000000000000)
     (- (* PI (fabs l)) (/ -1 (* (/ (- 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) <= 2.9e+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) <= 2.9e+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) <= 2.9e+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) <= 2.9e+15)
		tmp = Float64(Float64(pi * abs(l)) - Float64(-1.0 / Float64(Float64(Float64(-F) / 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) <= 2.9e+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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 2900000000000000], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(-1 / N[(N[((-F) / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9e15

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. *-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 81.7%

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

      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 81.7%

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

   3. Applied rewrites 81.7%

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

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

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

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

      4. lower-unsound-/.f64 81.7%

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 81.7%

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

   7. Applied rewrites 81.7%

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

   if 2.9e15 < l

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2900000000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1 l)
   (if (<= (fabs l) 2900000000000000)
     (- (* 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) <= 2.9e+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) <= 2.9e+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) <= 2.9e+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) <= 2.9e+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) <= 2.9e+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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 2900000000000000], 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 < 2.9e15

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. *-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 81.7%

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

      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 81.7%

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

   3. Applied rewrites 81.7%

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

   if 2.9e15 < l

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1300000000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{-1}{\left(-1 \cdot \frac{F}{t\_0}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1 l)
   (if (<= (fabs l) 1300000000000000)
     (- (* PI (fabs l)) (/ -1 (* (* -1 (/ F 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) <= 1.3e+15) {
		tmp = (((double) M_PI) * fabs(l)) - (-1.0 / ((-1.0 * (F / 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) <= 1.3e+15) {
		tmp = (Math.PI * Math.abs(l)) - (-1.0 / ((-1.0 * (F / 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) <= 1.3e+15:
		tmp = (math.pi * math.fabs(l)) - (-1.0 / ((-1.0 * (F / 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) <= 1.3e+15)
		tmp = Float64(Float64(pi * abs(l)) - Float64(-1.0 / Float64(Float64(-1.0 * Float64(F / 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) <= 1.3e+15)
		tmp = (pi * abs(l)) - (-1.0 / ((-1.0 * (F / 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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 1300000000000000], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(-1 / N[(N[(-1 * N[(F / t$95$0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. *-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 81.7%

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

      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 81.7%

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

   3. Applied rewrites 81.7%

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

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

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

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

      4. lower-unsound-/.f64 81.7%

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 81.7%

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 74.6%

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

   10. Applied rewrites 74.6%

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

   if 1.3e15 < l

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 4: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 1300000000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* (fabs l) PI)))
  (*
   (copysign 1 l)
   (if (<= (fabs l) 1300000000000000)
     (- (* 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) <= 1.3e+15) {
		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) <= 1.3e+15) {
		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) <= 1.3e+15:
		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) <= 1.3e+15)
		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) <= 1.3e+15)
		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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 1300000000000000], 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 < 1.3e15

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. *-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 81.7%

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

      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 81.7%

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

   3. Applied rewrites 81.7%

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

   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 74.6%

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

   6. Applied rewrites 74.6%

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

   if 1.3e15 < l

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 5: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq \frac{6176826577981891}{12353653155963782858428671064387042666598916611580939841119467653041402561523153496823753888027168243114445670429492914513538130943957121390808923336226715693451991227817771079374200911992994161452622538969283756626099089564495403968390097390695773702442020050803147555684412513136627351818013153603882218219464583806976}:\\ \;\;\;\;\frac{\left(\left(F \cdot F\right) \cdot \pi - \pi\right) \cdot \left|\ell\right|}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* PI (fabs l))))
  (*
   (copysign 1 l)
   (if (<=
        (- t_0 (* (/ 1 (* F F)) (tan t_0)))
        6176826577981891/12353653155963782858428671064387042666598916611580939841119467653041402561523153496823753888027168243114445670429492914513538130943957121390808923336226715693451991227817771079374200911992994161452622538969283756626099089564495403968390097390695773702442020050803147555684412513136627351818013153603882218219464583806976)
     (/ (* (- (* (* F F) PI) PI) (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))) <= 5e-304) {
		tmp = ((((F * F) * ((double) M_PI)) - ((double) M_PI)) * 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))) <= 5e-304) {
		tmp = ((((F * F) * Math.PI) - Math.PI) * 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))) <= 5e-304:
		tmp = ((((F * F) * math.pi) - math.pi) * 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))) <= 5e-304)
		tmp = Float64(Float64(Float64(Float64(Float64(F * F) * pi) - pi) * 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))) <= 5e-304)
		tmp = ((((F * F) * pi) - pi) * 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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$0 - N[(N[(1 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6176826577981891/12353653155963782858428671064387042666598916611580939841119467653041402561523153496823753888027168243114445670429492914513538130943957121390808923336226715693451991227817771079374200911992994161452622538969283756626099089564495403968390097390695773702442020050803147555684412513136627351818013153603882218219464583806976], N[(N[(N[(N[(N[(F * F), $MachinePrecision] * Pi), $MachinePrecision] - Pi), $MachinePrecision] * 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)))) < 4.9999999999999997e-304

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. sub-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-PI.f64 38.2%

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

   6. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.2%

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 38.2%

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

   8. Applied rewrites 38.2%

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

   if 4.9999999999999997e-304 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 6: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq \frac{-3974446316289815}{794889263257962974796277498092801308291525640763748664903194643469338087775424965801409745320266996710649718116931109481559848982586784968419475084821084743272680947722675151641735826243378403750534655587182832000457137589153821622272}:\\ \;\;\;\;\frac{\left|\ell\right| \cdot \left(-1 \cdot \pi\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
  :precision binary64
  (let* ((t_0 (* PI (fabs l))))
  (*
   (copysign 1 l)
   (if (<=
        (- t_0 (* (/ 1 (* F F)) (tan t_0)))
        -3974446316289815/794889263257962974796277498092801308291525640763748664903194643469338087775424965801409745320266996710649718116931109481559848982586784968419475084821084743272680947722675151641735826243378403750534655587182832000457137589153821622272)
     (/ (* (fabs l) (* -1 PI)) (* 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))) <= -5e-219) {
		tmp = (fabs(l) * (-1.0 * ((double) M_PI))) / (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))) <= -5e-219) {
		tmp = (Math.abs(l) * (-1.0 * Math.PI)) / (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))) <= -5e-219:
		tmp = (math.fabs(l) * (-1.0 * math.pi)) / (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))) <= -5e-219)
		tmp = Float64(Float64(abs(l) * Float64(-1.0 * pi)) / 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))) <= -5e-219)
		tmp = (abs(l) * (-1.0 * pi)) / (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], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$0 - N[(N[(1 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3974446316289815/794889263257962974796277498092801308291525640763748664903194643469338087775424965801409745320266996710649718116931109481559848982586784968419475084821084743272680947722675151641735826243378403750534655587182832000457137589153821622272], N[(N[(N[Abs[l], $MachinePrecision] * N[(-1 * Pi), $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)))) < -5.0000000000000002e-219

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. sub-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-PI.f64 38.2%

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

   6. Applied rewrites 38.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 20.6%

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

   9. Applied rewrites 20.6%

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

   if -5.0000000000000002e-219 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 74.2%

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

   4. Applied rewrites 74.2%

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

Alternative 7: 74.2% 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 75.9%

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

2. Taylor expanded in F around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 74.2%

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

4. Applied rewrites 74.2%

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

Alternative 8: 3.1% accurate, 22.5× speedup

Math

\[\ell \cdot 0 \]

FPCore

(FPCore (F l)
  :precision binary64
  (* l 0))

C

double code(double F, double l) {
	return l * 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, l)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: l
    code = l * 0.0d0
end function

Java

public static double code(double F, double l) {
	return l * 0.0;
}

Python

def code(F, l):
	return l * 0.0

Julia

function code(F, l)
	return Float64(l * 0.0)
end

MATLAB

function tmp = code(F, l)
	tmp = l * 0.0;
end

Wolfram

code[F_, l_] := N[(l * 0), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

2. Taylor expanded in F around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 74.2%

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

4. Applied rewrites 74.2%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \ell \cdot 0 \]
6. Step-by-step derivation

7. Applied rewrites 3.1%

   \[\leadsto \ell \cdot 0 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))