VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 98.5%

Time: 14.3s

Alternatives: 10

Speedup: 1.5×

• could not determine a ground truth

  1. F = 2.9456913492585937e+257
  2. l = 3007737837730684.5

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: 76.5% 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: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+38} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+38) (not (<= (* PI l) 10000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+38) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -1e+38) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e+38) or not ((math.pi * l) <= 10000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+38) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e+38) || ~(((pi * l) <= 10000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+38], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -9.99999999999999977e37 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 55.7%

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

      1. associate-*l/ 55.7%

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

      2. *-lft-identity 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in l around 0 41.2%

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

      1. unpow2 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -9.99999999999999977e37 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 89.4%

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

      1. associate-*l/ 91.1%

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

      2. *-un-lft-identity 91.1%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+38} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000 \lor \neg \left(\pi \cdot \ell \leq 50000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -10000.0) (not (<= (* PI l) 50000000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -10000.0) || !((((double) M_PI) * l) <= 50000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -10000.0) || !((Math.PI * l) <= 50000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -10000.0) or not ((math.pi * l) <= 50000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -10000.0) || !(Float64(pi * l) <= 50000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -10000.0) || ~(((pi * l) <= 50000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -10000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e4 or 5e7 < (*.f64 (PI.f64) l)

   1. Initial program 57.8%

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

      1. associate-*l/ 57.8%

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

      2. *-lft-identity 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 98.0%

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

   if -1e4 < (*.f64 (PI.f64) l) < 5e7

   1. Initial program 88.9%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 90.2%

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

      1. *-commutative 90.2%

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

      2. unpow2 90.2%

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

      3. times-frac 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000 \lor \neg \left(\pi \cdot \ell \leq 50000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1550000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ PI (/ F l)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / (F / l)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / (F / l)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1550000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / (F / l)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1550000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F / l)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1550000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / (F / l)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1550000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F / l), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.55e9 < l

   1. Initial program 57.8%

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

      1. associate-*l/ 57.8%

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

      2. *-lft-identity 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 98.0%

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

   if -0.5 < l < 1.55e9

   1. Initial program 88.9%

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

      1. associate-*l/ 90.7%

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

      2. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 88.4%

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

      1. unpow2 88.4%

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

   6. Simplified 88.4%

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

      1. *-un-lft-identity 88.4%

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

      2. times-frac 88.4%

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

   8. Applied egg-rr 88.4%

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

      1. associate-*l/ 88.4%

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

      2. *-lft-identity 88.4%

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

   10. Simplified 88.4%

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

       1. clear-num 88.4%

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

       2. associate-/r/ 88.4%

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

   12. Applied egg-rr 88.4%

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

   13. Taylor expanded in F around 0 90.2%

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

       1. mul-1-neg 90.2%

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

       2. unsub-neg 90.2%

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

       3. unpow2 90.2%

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

       4. times-frac 99.2%

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

       5. associate-*l/ 99.0%

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

       6. *-commutative 99.0%

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

       7. associate-*l/ 99.1%

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

       8. associate-/l* 99.1%

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

   15. Simplified 99.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \end{array} \]

Alternative 4: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1550000000.0)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1550000000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1550000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1550000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1550000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.55e9 < l

   1. Initial program 57.8%

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

      1. associate-*l/ 57.8%

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

      2. *-lft-identity 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 98.0%

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

   if -0.5 < l < 1.55e9

   1. Initial program 88.9%

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

      1. associate-*l/ 90.7%

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

      2. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 88.4%

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

      1. unpow2 88.4%

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

   6. Simplified 88.4%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\frac{\pi}{F}}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1550000000.0)))
   (* PI l)
   (* l (- PI (/ (/ PI F) F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - ((((double) M_PI) / F) / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1550000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - ((Math.PI / F) / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1550000000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - ((math.pi / F) / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1550000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(Float64(pi / F) / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1550000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - ((pi / F) / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1550000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(N[(Pi / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.55e9 < l

   1. Initial program 57.8%

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

      1. associate-*l/ 57.8%

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

      2. *-lft-identity 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 98.0%

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

   if -0.5 < l < 1.55e9

   1. Initial program 88.9%

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

      1. associate-*l/ 90.7%

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

      2. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 88.4%

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

      1. unpow2 88.4%

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

   6. Simplified 88.4%

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

      1. *-un-lft-identity 88.4%

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

      2. times-frac 88.4%

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

   8. Applied egg-rr 88.4%

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

      1. associate-*l/ 88.4%

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

      2. *-lft-identity 88.4%

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

   10. Simplified 88.4%

       \[\leadsto \ell \cdot \left(\pi - \color{blue}{\frac{\frac{\pi}{F}}{F}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1550000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\frac{\pi}{F}}{F}\right)\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1420000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1420000000.0)))
   (* PI l)
   (* (* PI l) (- 1.0 (pow F -2.0)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1420000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) * (1.0 - pow(F, -2.0));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1420000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) * (1.0 - Math.pow(F, -2.0));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1420000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) * (1.0 - math.pow(F, -2.0))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1420000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) * Float64(1.0 - (F ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1420000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) * (1.0 - (F ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1420000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.42e9 < l

   1. Initial program 57.8%

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

      1. associate-*l/ 57.8%

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

      2. *-lft-identity 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 98.0%

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

   if -0.5 < l < 1.42e9

   1. Initial program 88.9%

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

      1. associate-*l/ 90.7%

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

      2. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 88.4%

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

      1. unpow2 88.4%

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

   6. Simplified 88.4%

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

      1. *-un-lft-identity 88.4%

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

      2. times-frac 88.4%

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

   8. Applied egg-rr 88.4%

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

      1. associate-*l/ 88.4%

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

      2. *-lft-identity 88.4%

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

   10. Simplified 88.4%

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

   11. Taylor expanded in l around 0 88.4%

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

       1. sub-neg 88.4%

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

       2. distribute-lft-in 88.4%

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

       3. unpow2 88.4%

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

       4. *-rgt-identity 88.4%

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

       5. times-frac 88.4%

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

       6. *-rgt-identity 88.4%

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

       7. associate-*r/ 88.3%

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

       8. associate-*r* 88.3%

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

       9. unpow-1 88.3%

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

       10. unpow-1 88.3%

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

       11. pow-sqr 88.4%

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

       12. metadata-eval 88.4%

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

       13. distribute-rgt-neg-out 88.4%

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

       14. *-rgt-identity 88.4%

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

       15. associate-*r* 88.4%

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

   13. Simplified 88.4%

       \[\leadsto \color{blue}{\left(\ell \cdot \pi\right) \cdot \left(1 - {F}^{-2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1420000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \end{array} \]

Alternative 7: 75.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-261} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-207}\right) \land \left(F \cdot F \leq 2 \cdot 10^{-68} \lor \neg \left(F \cdot F \leq 5 \cdot 10^{-14}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* F F) 4e-261)
         (and (not (<= (* F F) 2e-207))
              (or (<= (* F F) 2e-68) (not (<= (* F F) 5e-14)))))
   (* PI l)
   (* (/ PI (* F F)) (- l))))

C

double code(double F, double l) {
	double tmp;
	if (((F * F) <= 4e-261) || (!((F * F) <= 2e-207) && (((F * F) <= 2e-68) || !((F * F) <= 5e-14)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) / (F * F)) * -l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((F * F) <= 4e-261) || (!((F * F) <= 2e-207) && (((F * F) <= 2e-68) || !((F * F) <= 5e-14)))) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI / (F * F)) * -l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((F * F) <= 4e-261) or (not ((F * F) <= 2e-207) and (((F * F) <= 2e-68) or not ((F * F) <= 5e-14))):
		tmp = math.pi * l
	else:
		tmp = (math.pi / (F * F)) * -l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(F * F) <= 4e-261) || (!(Float64(F * F) <= 2e-207) && ((Float64(F * F) <= 2e-68) || !(Float64(F * F) <= 5e-14))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi / Float64(F * F)) * Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((F * F) <= 4e-261) || (~(((F * F) <= 2e-207)) && (((F * F) <= 2e-68) || ~(((F * F) <= 5e-14)))))
		tmp = pi * l;
	else
		tmp = (pi / (F * F)) * -l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(F * F), $MachinePrecision], 4e-261], And[N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-207]], $MachinePrecision], Or[LessEqual[N[(F * F), $MachinePrecision], 2e-68], N[Not[LessEqual[N[(F * F), $MachinePrecision], 5e-14]], $MachinePrecision]]]], N[(Pi * l), $MachinePrecision], N[(N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision] * (-l)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 3.99999999999999994e-261 or 1.99999999999999985e-207 < (*.f64 F F) < 2.00000000000000013e-68 or 5.0000000000000002e-14 < (*.f64 F F)

   1. Initial program 72.2%

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

      1. associate-*l/ 73.2%

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

      2. *-lft-identity 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in l around 0 64.6%

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

      1. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in F around inf 80.0%

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

   if 3.99999999999999994e-261 < (*.f64 F F) < 1.99999999999999985e-207 or 2.00000000000000013e-68 < (*.f64 F F) < 5.0000000000000002e-14

   1. Initial program 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in l around 0 78.5%

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

      1. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in F around 0 78.0%

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

      1. unpow2 78.0%

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

      2. associate-*r/ 78.0%

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

      3. neg-mul-1 78.0%

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

   9. Simplified 78.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-261} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-207}\right) \land \left(F \cdot F \leq 2 \cdot 10^{-68} \lor \neg \left(F \cdot F \leq 5 \cdot 10^{-14}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \end{array} \]

Alternative 8: 75.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-261}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-207}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-68} \lor \neg \left(F \cdot F \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{-F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 4e-261)
   (* PI l)
   (if (<= (* F F) 2e-207)
     (* (/ PI (* F F)) (- l))
     (if (or (<= (* F F) 2e-68) (not (<= (* F F) 5e-14)))
       (* PI l)
       (* l (/ (/ PI (- F)) F))))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 4e-261) {
		tmp = ((double) M_PI) * l;
	} else if ((F * F) <= 2e-207) {
		tmp = (((double) M_PI) / (F * F)) * -l;
	} else if (((F * F) <= 2e-68) || !((F * F) <= 5e-14)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * ((((double) M_PI) / -F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 4e-261) {
		tmp = Math.PI * l;
	} else if ((F * F) <= 2e-207) {
		tmp = (Math.PI / (F * F)) * -l;
	} else if (((F * F) <= 2e-68) || !((F * F) <= 5e-14)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * ((Math.PI / -F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 4e-261:
		tmp = math.pi * l
	elif (F * F) <= 2e-207:
		tmp = (math.pi / (F * F)) * -l
	elif ((F * F) <= 2e-68) or not ((F * F) <= 5e-14):
		tmp = math.pi * l
	else:
		tmp = l * ((math.pi / -F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 4e-261)
		tmp = Float64(pi * l);
	elseif (Float64(F * F) <= 2e-207)
		tmp = Float64(Float64(pi / Float64(F * F)) * Float64(-l));
	elseif ((Float64(F * F) <= 2e-68) || !(Float64(F * F) <= 5e-14))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(Float64(pi / Float64(-F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 4e-261)
		tmp = pi * l;
	elseif ((F * F) <= 2e-207)
		tmp = (pi / (F * F)) * -l;
	elseif (((F * F) <= 2e-68) || ~(((F * F) <= 5e-14)))
		tmp = pi * l;
	else
		tmp = l * ((pi / -F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 4e-261], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 2e-207], N[(N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision] * (-l)), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 2e-68], N[Not[LessEqual[N[(F * F), $MachinePrecision], 5e-14]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(N[(Pi / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 3.99999999999999994e-261 or 1.99999999999999985e-207 < (*.f64 F F) < 2.00000000000000013e-68 or 5.0000000000000002e-14 < (*.f64 F F)

   1. Initial program 72.2%

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

      1. associate-*l/ 73.2%

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

      2. *-lft-identity 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in l around 0 64.6%

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

      1. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in F around inf 80.0%

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

   if 3.99999999999999994e-261 < (*.f64 F F) < 1.99999999999999985e-207

   1. Initial program 82.3%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. 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. *-lft-identity 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in l around 0 80.9%

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

      1. unpow2 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in F around 0 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r/ 80.9%

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

      3. neg-mul-1 80.9%

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

   9. Simplified 80.9%

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

   if 2.00000000000000013e-68 < (*.f64 F F) < 5.0000000000000002e-14

   1. Initial program 93.9%

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

      1. associate-*l/ 93.8%

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

      2. *-lft-identity 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in l around 0 76.5%

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

      1. unpow2 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in F around 0 75.5%

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

      1. unpow2 75.5%

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

      2. associate-*r/ 75.5%

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

      3. neg-mul-1 75.5%

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

   9. Simplified 75.5%

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

   10. Taylor expanded in F around 0 75.5%

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

       1. metadata-eval 75.5%

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

       2. unpow2 75.5%

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

       3. times-frac 75.5%

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

       4. *-lft-identity 75.5%

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

       5. neg-mul-1 75.5%

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

       6. distribute-lft-neg-in 75.5%

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

       7. associate-/r* 75.8%

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

   12. Simplified 75.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-261}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-207}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-68} \lor \neg \left(F \cdot F \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{-F}}{F}\\ \end{array} \]

Alternative 9: 75.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{-F}\\ \mathbf{if}\;F \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\frac{\ell}{F} \cdot t_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-34} \lor \neg \left(F \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{t_0}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (/ PI (- F))))
   (if (<= F 7e-287)
     (* PI l)
     (if (<= F 6e-103)
       (* (/ l F) t_0)
       (if (or (<= F 5.8e-34) (not (<= F 4.6e-7)))
         (* PI l)
         (* l (/ t_0 F)))))))

C

double code(double F, double l) {
	double t_0 = ((double) M_PI) / -F;
	double tmp;
	if (F <= 7e-287) {
		tmp = ((double) M_PI) * l;
	} else if (F <= 6e-103) {
		tmp = (l / F) * t_0;
	} else if ((F <= 5.8e-34) || !(F <= 4.6e-7)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (t_0 / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.PI / -F;
	double tmp;
	if (F <= 7e-287) {
		tmp = Math.PI * l;
	} else if (F <= 6e-103) {
		tmp = (l / F) * t_0;
	} else if ((F <= 5.8e-34) || !(F <= 4.6e-7)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (t_0 / F);
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = math.pi / -F
	tmp = 0
	if F <= 7e-287:
		tmp = math.pi * l
	elif F <= 6e-103:
		tmp = (l / F) * t_0
	elif (F <= 5.8e-34) or not (F <= 4.6e-7):
		tmp = math.pi * l
	else:
		tmp = l * (t_0 / F)
	return tmp

Julia

function code(F, l)
	t_0 = Float64(pi / Float64(-F))
	tmp = 0.0
	if (F <= 7e-287)
		tmp = Float64(pi * l);
	elseif (F <= 6e-103)
		tmp = Float64(Float64(l / F) * t_0);
	elseif ((F <= 5.8e-34) || !(F <= 4.6e-7))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(t_0 / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = pi / -F;
	tmp = 0.0;
	if (F <= 7e-287)
		tmp = pi * l;
	elseif (F <= 6e-103)
		tmp = (l / F) * t_0;
	elseif ((F <= 5.8e-34) || ~((F <= 4.6e-7)))
		tmp = pi * l;
	else
		tmp = l * (t_0 / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(Pi / (-F)), $MachinePrecision]}, If[LessEqual[F, 7e-287], N[(Pi * l), $MachinePrecision], If[LessEqual[F, 6e-103], N[(N[(l / F), $MachinePrecision] * t$95$0), $MachinePrecision], If[Or[LessEqual[F, 5.8e-34], N[Not[LessEqual[F, 4.6e-7]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 7e-287 or 6e-103 < F < 5.8000000000000004e-34 or 4.5999999999999999e-7 < F

   1. Initial program 79.9%

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

      1. associate-*l/ 80.7%

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

      2. *-lft-identity 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in l around 0 71.8%

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

      1. unpow2 71.8%

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

   6. Simplified 71.8%

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

   7. Taylor expanded in F around inf 83.9%

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

   if 7e-287 < F < 6e-103

   1. Initial program 43.7%

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

      1. associate-*l/ 45.3%

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

      2. *-lft-identity 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in l around 0 38.7%

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

      1. unpow2 38.7%

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

   6. Simplified 38.7%

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

   7. Taylor expanded in F around 0 38.7%

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

      1. unpow2 38.7%

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

      2. associate-*r/ 38.7%

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

      3. neg-mul-1 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in l around 0 40.3%

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

       1. mul-1-neg 40.3%

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

       2. unpow2 40.3%

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

       3. times-frac 55.1%

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

       4. distribute-rgt-neg-in 55.1%

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

       5. neg-mul-1 55.1%

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

       6. metadata-eval 55.1%

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

       7. times-frac 55.1%

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

       8. *-lft-identity 55.1%

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

       9. neg-mul-1 55.1%

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

       10. times-frac 40.3%

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

       11. *-commutative 40.3%

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

       12. distribute-rgt-neg-in 40.3%

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

       13. distribute-lft-neg-in 40.3%

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

       14. times-frac 55.1%

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

   12. Simplified 55.1%

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

   if 5.8000000000000004e-34 < F < 4.5999999999999999e-7

   1. Initial program 90.3%

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

      1. associate-*l/ 90.3%

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

      2. *-lft-identity 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in l around 0 74.7%

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

      1. unpow2 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in F around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r/ 74.7%

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

      3. neg-mul-1 74.7%

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

   9. Simplified 74.7%

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

   10. Taylor expanded in F around 0 74.7%

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

       1. metadata-eval 74.7%

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

       2. unpow2 74.7%

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

       3. times-frac 74.7%

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

       4. *-lft-identity 74.7%

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

       5. neg-mul-1 74.7%

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

       6. distribute-lft-neg-in 74.7%

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

       7. associate-/r* 75.0%

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

   12. Simplified 75.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-34} \lor \neg \left(F \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{-F}}{F}\\ \end{array} \]

Alternative 10: 74.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell \end{array} \]

FPCore

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

C

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

Java

public static double code(double F, double l) {
	return Math.PI * l;
}

Python

def code(F, l):
	return math.pi * l

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l/ 74.7%

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

   2. *-lft-identity 74.7%

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

3. Simplified 74.7%

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

4. Taylor expanded in l around 0 65.9%

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

   1. unpow2 65.9%

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

6. Simplified 65.9%

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

7. Taylor expanded in F around inf 74.6%

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

8. Final simplification 74.6%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))