VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.6% → 98.8%

Time: 3.4s

Alternatives: 5

Speedup: 2.2×

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: 75.6% 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: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 2.9e+21) (- (* 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+21) {
		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+21) {
		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+21:
		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+21)
		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+21)
		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], 2.9e+21], 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 < 2.9e21

   1. Initial program 75.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.4

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

      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 81.4

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

   3. Applied rewrites 81.4%

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

   if 2.9e21 < l

   1. Initial program 75.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.9

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

   4. Applied rewrites 73.9%

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

Alternative 2: 98.4% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2400000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\pi}{F} \cdot \left|\ell\right|}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 2400000000.0)
    (- (* PI (fabs l)) (/ (* (/ PI F) (fabs l)) F))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 2400000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - (((((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 tmp;
	if (Math.abs(l) <= 2400000000.0) {
		tmp = (Math.PI * Math.abs(l)) - (((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):
	tmp = 0
	if math.fabs(l) <= 2400000000.0:
		tmp = (math.pi * math.fabs(l)) - (((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)
	tmp = 0.0
	if (abs(l) <= 2400000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(Float64(pi / F) * 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)
	tmp = 0.0;
	if (abs(l) <= 2400000000.0)
		tmp = (pi * abs(l)) - (((pi / F) * abs(l)) / 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], 2400000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(Pi / F), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.4e9

   1. Initial program 75.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.4

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

      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 81.4

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

   3. Applied rewrites 81.4%

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

   4. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   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 74.0

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

   6. Applied rewrites 74.0%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if 2.4e9 < l

   1. Initial program 75.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.9

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

   4. Applied rewrites 73.9%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2400000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\pi \cdot \frac{\left|\ell\right|}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 2400000000.0)
    (- (* PI (fabs l)) (/ (* PI (/ (fabs l) F)) F))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 2400000000.0) {
		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) <= 2400000000.0) {
		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) <= 2400000000.0:
		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) <= 2400000000.0)
		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) <= 2400000000.0)
		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], 2400000000.0], 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 < 2.4e9

   1. Initial program 75.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.4

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

      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 81.4

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

   3. Applied rewrites 81.4%

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

   4. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   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 74.0

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

   6. Applied rewrites 74.0%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
   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 74.0

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

   8. Applied rewrites 74.0%

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

   if 2.4e9 < l

   1. Initial program 75.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.9

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

   4. Applied rewrites 73.9%

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

Alternative 4: 92.6% accurate, 2.2× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2400000000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot \left|\ell\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 2400000000.0)
    (* (- PI (/ PI (* F F))) (fabs l))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 2400000000.0) {
		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) <= 2400000000.0) {
		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) <= 2400000000.0:
		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) <= 2400000000.0)
		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) <= 2400000000.0)
		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], 2400000000.0], 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 < 2.4e9

   1. Initial program 75.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 58.0%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-pow.f64 38.3

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

   6. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. sub-to-fraction-rev N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 68.2

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if 2.4e9 < l

   1. Initial program 75.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.9

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

   4. Applied rewrites 73.9%

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

Alternative 5: 73.9% 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 75.6%

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

2. Taylor expanded in F around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 73.9

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

4. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2025178 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))