VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.1%

Time: 20.1s

Alternatives: 5

Speedup: 6.3×

• could not determine a ground truth

  1. F = 3.262065688143196e+170
  2. l = -776391013804492.5

Specification

Math

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

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: 76.9% accurate, 1.0× speedup

Math

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

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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 100000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 100000000000.0)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 100000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 100000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 100000000000.0:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 100000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 100000000000.0)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 100000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e11

   1. Initial program 81.5%

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

      1. associate-*l/ 82.3%

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

      2. *-un-lft-identity 82.3%

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

      3. associate-/r* 88.4%

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

   4. Applied egg-rr 88.4%

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

   if 1e11 < (*.f64 (PI.f64) l)

   1. Initial program 64.1%

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

   3. Taylor expanded in l around inf 99.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 100000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-281}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{F}{l\_m}}}{-F}\\ \mathbf{elif}\;\pi \cdot l\_m \leq 5 \cdot 10^{-237} \lor \neg \left(\pi \cdot l\_m \leq 4 \cdot 10^{-104}\right):\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{l\_m}{F}\right)\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-281)
    (/ (/ PI (/ F l_m)) (- F))
    (if (or (<= (* PI l_m) 5e-237) (not (<= (* PI l_m) 4e-104)))
      (* PI l_m)
      (* (/ PI F) (- (/ l_m F)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-281) {
		tmp = (((double) M_PI) / (F / l_m)) / -F;
	} else if (((((double) M_PI) * l_m) <= 5e-237) || !((((double) M_PI) * l_m) <= 4e-104)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = (((double) M_PI) / F) * -(l_m / F);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e-281) {
		tmp = (Math.PI / (F / l_m)) / -F;
	} else if (((Math.PI * l_m) <= 5e-237) || !((Math.PI * l_m) <= 4e-104)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = (Math.PI / F) * -(l_m / F);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e-281:
		tmp = (math.pi / (F / l_m)) / -F
	elif ((math.pi * l_m) <= 5e-237) or not ((math.pi * l_m) <= 4e-104):
		tmp = math.pi * l_m
	else:
		tmp = (math.pi / F) * -(l_m / F)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-281)
		tmp = Float64(Float64(pi / Float64(F / l_m)) / Float64(-F));
	elseif ((Float64(pi * l_m) <= 5e-237) || !(Float64(pi * l_m) <= 4e-104))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(pi / F) * Float64(-Float64(l_m / F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e-281)
		tmp = (pi / (F / l_m)) / -F;
	elseif (((pi * l_m) <= 5e-237) || ~(((pi * l_m) <= 4e-104)))
		tmp = pi * l_m;
	else
		tmp = (pi / F) * -(l_m / F);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-281], N[(N[(Pi / N[(F / l$95$m), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision], If[Or[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e-237], N[Not[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 4e-104]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(Pi / F), $MachinePrecision] * (-N[(l$95$m / F), $MachinePrecision])), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 1e-281

   1. Initial program 75.9%

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

   3. Taylor expanded in l around 0 68.6%

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

   4. Taylor expanded in F around 0 24.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 24.6%

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

      2. associate-/l* 23.9%

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

   6. Simplified 23.9%

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

      1. associate-*r/ 24.6%

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

      2. *-commutative 24.6%

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

      3. unpow2 24.6%

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

      4. frac-times 30.2%

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

      5. associate-*l/ 30.1%

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

      6. distribute-neg-frac2 30.1%

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

      7. clear-num 30.1%

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

      8. un-div-inv 30.2%

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

   8. Applied egg-rr 30.2%

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

   if 1e-281 < (*.f64 (PI.f64) l) < 5.0000000000000002e-237 or 3.99999999999999971e-104 < (*.f64 (PI.f64) l)

   1. Initial program 74.4%

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

   3. Taylor expanded in l around inf 88.7%

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

   if 5.0000000000000002e-237 < (*.f64 (PI.f64) l) < 3.99999999999999971e-104

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0 87.0%

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

   4. Taylor expanded in F around 0 54.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 54.3%

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

      2. associate-/l* 54.3%

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

   6. Simplified 54.3%

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

      1. associate-*r/ 54.3%

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

      2. *-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. frac-times 66.8%

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

      5. distribute-rgt-neg-in 66.8%

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

   8. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-281}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{F}{\ell}}}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-237} \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-104}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-281} \lor \neg \left(\pi \cdot l\_m \leq 5 \cdot 10^{-237}\right) \land \pi \cdot l\_m \leq 4 \cdot 10^{-104}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{l\_m}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= (* PI l_m) 1e-281)
          (and (not (<= (* PI l_m) 5e-237)) (<= (* PI l_m) 4e-104)))
    (* (/ PI F) (- (/ l_m F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (((((double) M_PI) * l_m) <= 1e-281) || (!((((double) M_PI) * l_m) <= 5e-237) && ((((double) M_PI) * l_m) <= 4e-104))) {
		tmp = (((double) M_PI) / F) * -(l_m / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (((Math.PI * l_m) <= 1e-281) || (!((Math.PI * l_m) <= 5e-237) && ((Math.PI * l_m) <= 4e-104))) {
		tmp = (Math.PI / F) * -(l_m / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if ((math.pi * l_m) <= 1e-281) or (not ((math.pi * l_m) <= 5e-237) and ((math.pi * l_m) <= 4e-104)):
		tmp = (math.pi / F) * -(l_m / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((Float64(pi * l_m) <= 1e-281) || (!(Float64(pi * l_m) <= 5e-237) && (Float64(pi * l_m) <= 4e-104)))
		tmp = Float64(Float64(pi / F) * Float64(-Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (((pi * l_m) <= 1e-281) || (~(((pi * l_m) <= 5e-237)) && ((pi * l_m) <= 4e-104)))
		tmp = (pi / F) * -(l_m / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[Or[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-281], And[N[Not[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e-237]], $MachinePrecision], LessEqual[N[(Pi * l$95$m), $MachinePrecision], 4e-104]]], N[(N[(Pi / F), $MachinePrecision] * (-N[(l$95$m / F), $MachinePrecision])), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-281 or 5.0000000000000002e-237 < (*.f64 (PI.f64) l) < 3.99999999999999971e-104

   1. Initial program 78.1%

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

   3. Taylor expanded in l around 0 72.3%

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

   4. Taylor expanded in F around 0 30.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 30.5%

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

      2. associate-/l* 29.9%

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

   6. Simplified 29.9%

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

      1. associate-*r/ 30.5%

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

      2. *-commutative 30.5%

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

      3. unpow2 30.5%

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

      4. frac-times 37.5%

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

      5. distribute-rgt-neg-in 37.5%

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

   8. Applied egg-rr 37.5%

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

   if 1e-281 < (*.f64 (PI.f64) l) < 5.0000000000000002e-237 or 3.99999999999999971e-104 < (*.f64 (PI.f64) l)

   1. Initial program 74.4%

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

   3. Taylor expanded in l around inf 88.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-281} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-237}\right) \land \pi \cdot \ell \leq 4 \cdot 10^{-104}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 100000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 100000000000.0)
    (- (* PI l_m) (* (/ PI F) (/ l_m F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 100000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 100000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 100000000000.0:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 100000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 100000000000.0)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 100000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e11

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. pow2 76.9%

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

      3. times-frac 83.1%

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

   5. Applied egg-rr 83.1%

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

   if 1e11 < (*.f64 (PI.f64) l)

   1. Initial program 64.1%

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

   3. Taylor expanded in l around inf 99.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 100000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 37.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* PI l_m)))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * (((double) M_PI) * l_m);
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * (Math.PI * l_m);
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * (math.pi * l_m)

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(pi * l_m))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * (pi * l_m);
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in l around inf 74.7%

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

4. Final simplification 74.7%

   \[\leadsto \pi \cdot \ell \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))