VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.5% → 99.2%

Time: 3.2s

Alternatives: 5

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: 75.5% 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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3000000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 3000000000000.0)
      (- (* 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) <= 3000000000000.0) {
		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) <= 3000000000000.0) {
		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) <= 3000000000000.0:
		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) <= 3000000000000.0)
		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) <= 3000000000000.0)
		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], 3000000000000.0], 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 < 3e12

   1. Initial program 75.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.5%

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

      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 81.5%

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

   3. Applied rewrites 81.5%

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

   if 3e12 < l

   1. Initial program 75.5%

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

   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 73.9%

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

   4. Applied rewrites 73.9%

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

Alternative 2: 98.1% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 0.0007:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \pi \cdot \frac{\frac{\left|\ell\right|}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 0.0007)
    (- (* PI (fabs l)) (* PI (/ (/ (fabs l) F) F)))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 0.0007) {
		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) <= 0.0007) {
		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) <= 0.0007:
		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) <= 0.0007)
		tmp = Float64(Float64(pi * abs(l)) - Float64(pi * Float64(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) <= 0.0007)
		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], 0.0007], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(Pi * N[(N[(N[Abs[l], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.99999999999999993e-4

   1. Initial program 75.5%

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

   2. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-pow.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 68.7%

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 68.7%

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

   6. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 74.3%

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

   8. Applied rewrites 74.3%

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

   if 6.99999999999999993e-4 < l

   1. Initial program 75.5%

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

   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 73.9%

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

   4. Applied rewrites 73.9%

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

Alternative 3: 98.0% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 0.0007:\\ \;\;\;\;\frac{\left|\ell\right| \cdot \left(F \cdot \pi - \frac{\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 0.0007)
    (/ (* (fabs l) (- (* F PI) (/ PI F))) F)
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 0.0007) {
		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) <= 0.0007) {
		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) <= 0.0007:
		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) <= 0.0007)
		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) <= 0.0007)
		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], 0.0007], 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 < 6.99999999999999993e-4

   1. Initial program 75.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 81.5%

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

      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 81.5%

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

   3. Applied rewrites 81.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\ell \cdot \pi\right)}{F}}{F}} \]
   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 70.0%

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

      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 70.0%

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

   5. Applied rewrites 70.0%

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

   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 62.8%

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

   8. Applied rewrites 62.8%

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

   if 6.99999999999999993e-4 < l

   1. Initial program 75.5%

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

   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 73.9%

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

   4. Applied rewrites 73.9%

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

Alternative 4: 92.5% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 0.0007:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \pi \cdot \frac{\left|\ell\right|}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 0.0007)
    (- (* PI (fabs l)) (* PI (/ (fabs l) (* F F))))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 0.0007) {
		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) <= 0.0007) {
		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) <= 0.0007:
		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) <= 0.0007)
		tmp = Float64(Float64(pi * abs(l)) - Float64(pi * Float64(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)
	tmp = 0.0;
	if (abs(l) <= 0.0007)
		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], 0.0007], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(Pi * N[(N[Abs[l], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.99999999999999993e-4

   1. Initial program 75.5%

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

   2. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-pow.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 68.7%

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 68.7%

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

   6. Applied rewrites 68.7%

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

   if 6.99999999999999993e-4 < l

   1. Initial program 75.5%

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

   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 73.9%

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

   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 75.5%

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

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 73.9%

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

4. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2025185 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))