VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.7% → 99.3%

Time: 3.5s

Alternatives: 6

Speedup: 1.9×

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) (* (/ 1.0 (* 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: 76.7% 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

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{1}{\frac{F}{\tan t\_0}}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 9e+14)
      (- (* 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) <= 9e+14) {
		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) <= 9e+14) {
		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) <= 9e+14:
		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) <= 9e+14)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(1.0 / 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) <= 9e+14)
		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], 9e+14], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(F / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9e14

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.4%

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

   3. Applied rewrites 82.4%

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-unsound-/.f64 82.4%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 82.4%

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

   5. Applied rewrites 82.4%

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

   if 9e14 < l

   1. Initial program 76.7%

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

   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 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 9e+14) (- (* 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) <= 9e+14) {
		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) <= 9e+14) {
		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) <= 9e+14:
		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) <= 9e+14)
		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) <= 9e+14)
		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], 9e+14], 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 < 9e14

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.4%

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

   3. Applied rewrites 82.4%

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

   if 9e14 < l

   1. Initial program 76.7%

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

   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 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 720000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\pi \cdot \frac{\left|\ell\right|}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 720000000000.0)
    (- (* PI (fabs l)) (/ (* PI (/ (fabs l) F)) F))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 720000000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - ((((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 tmp;
	if (Math.abs(l) <= 720000000000.0) {
		tmp = (Math.PI * Math.abs(l)) - ((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):
	tmp = 0
	if math.fabs(l) <= 720000000000.0:
		tmp = (math.pi * math.fabs(l)) - ((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)
	tmp = 0.0
	if (abs(l) <= 720000000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(pi * Float64(abs(l) / F)) / F));
	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) <= 720000000000.0)
		tmp = (pi * abs(l)) - ((pi * (abs(l) / F)) / F);
	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], 720000000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * N[(N[Abs[l], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.2e11

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.4%

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

   3. Applied rewrites 82.4%

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

   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 74.8%

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

   6. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.9%

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

   8. Applied rewrites 74.9%

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

   if 7.2e11 < l

   1. Initial program 76.7%

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

   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 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 560000000000:\\ \;\;\;\;\frac{\left|\ell\right| \cdot \left(F \cdot \pi - \frac{\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 560000000000.0)
    (/ (* (fabs l) (- (* F PI) (/ PI F))) F)
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 560000000000.0) {
		tmp = (fabs(l) * ((F * ((double) M_PI)) - (((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 tmp;
	if (Math.abs(l) <= 560000000000.0) {
		tmp = (Math.abs(l) * ((F * Math.PI) - (Math.PI / F))) / F;
	} 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) <= 560000000000.0:
		tmp = (math.fabs(l) * ((F * math.pi) - (math.pi / F))) / F
	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) <= 560000000000.0)
		tmp = Float64(Float64(abs(l) * Float64(Float64(F * pi) - Float64(pi / F))) / F);
	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) <= 560000000000.0)
		tmp = (abs(l) * ((F * pi) - (pi / F))) / F;
	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], 560000000000.0], N[(N[(N[Abs[l], $MachinePrecision] * N[(N[(F * Pi), $MachinePrecision] - N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.6e11

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.4%

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

   3. Applied rewrites 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.7%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 71.7%

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 64.2%

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

   8. Applied rewrites 64.2%

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

   if 5.6e11 < l

   1. Initial program 76.7%

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

   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 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ t_1 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 - \frac{1}{F \cdot F} \cdot \tan t\_1 \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1 \cdot t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)) (t_1 (* PI (fabs l))))
   (*
    (copysign 1.0 l)
    (if (<= (- t_1 (* (/ 1.0 (* F F)) (tan t_1))) -1e-280)
      (/ (* -1.0 t_0) (* F F))
      t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double t_1 = ((double) M_PI) * fabs(l);
	double tmp;
	if ((t_1 - ((1.0 / (F * F)) * tan(t_1))) <= -1e-280) {
		tmp = (-1.0 * 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 t_1 = Math.PI * Math.abs(l);
	double tmp;
	if ((t_1 - ((1.0 / (F * F)) * Math.tan(t_1))) <= -1e-280) {
		tmp = (-1.0 * 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
	t_1 = math.pi * math.fabs(l)
	tmp = 0
	if (t_1 - ((1.0 / (F * F)) * math.tan(t_1))) <= -1e-280:
		tmp = (-1.0 * 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)
	t_1 = Float64(pi * abs(l))
	tmp = 0.0
	if (Float64(t_1 - Float64(Float64(1.0 / Float64(F * F)) * tan(t_1))) <= -1e-280)
		tmp = Float64(Float64(-1.0 * t_0) / Float64(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;
	t_1 = pi * abs(l);
	tmp = 0.0;
	if ((t_1 - ((1.0 / (F * F)) * tan(t_1))) <= -1e-280)
		tmp = (-1.0 * 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]}, Block[{t$95$1 = 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$1 - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-280], N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], t$95$0]), $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)))) < -9.9999999999999996e-281

   1. Initial program 76.7%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. distribute-neg-frac N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 23.4%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 21.8%

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

   9. Applied rewrites 21.8%

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

   if -9.9999999999999996e-281 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.7%

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

   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 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 6: 73.1% accurate, 13.6× 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 76.7%

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

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 73.1%

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

4. Applied rewrites 73.1%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))