VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 98.5%

Time: 26.7s

Alternatives: 8

Speedup: 1.5×

• could not determine a ground truth

  1. F = 3.0849770424215736e+224
  2. l = 3502908275653060.5

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: 76.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+34} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+34) (not (<= (* PI l) 2e+15)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+34) || !((((double) M_PI) * l) <= 2e+15)) {
		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+34) || !((Math.PI * l) <= 2e+15)) {
		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+34) or not ((math.pi * l) <= 2e+15):
		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+34) || !(Float64(pi * l) <= 2e+15))
		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+34) || ~(((pi * l) <= 2e+15)))
		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+34], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e+15]], $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) < -9.99999999999999946e33 or 2e15 < (*.f64 (PI.f64) l)

   1. Initial program 62.6%

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

      1. sqr-neg 62.6%

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

      2. associate-*l/ 62.6%

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

      3. *-lft-identity 62.6%

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

      4. sqr-neg 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in l around 0 47.1%

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

      1. unpow2 47.1%

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

   6. Simplified 47.1%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -9.99999999999999946e33 < (*.f64 (PI.f64) l) < 2e15

   1. Initial program 90.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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* 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+34} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -40000.0) (not (<= (* PI l) 2e+15)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -40000.0) || !((((double) M_PI) * l) <= 2e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((((double) M_PI) * l) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -40000.0) || !((Math.PI * l) <= 2e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (((Math.PI * l) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -40000.0) or not ((math.pi * l) <= 2e+15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (((math.pi * l) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -40000.0) || !(Float64(pi * l) <= 2e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -40000.0) || ~(((pi * l) <= 2e+15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (((pi * l) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -40000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -4e4 or 2e15 < (*.f64 (PI.f64) l)

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -4e4 < (*.f64 (PI.f64) l) < 2e15

   1. Initial program 90.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in l around 0 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 3: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1.5e+15)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1.5e+15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1.5e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1.5e+15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1.5e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.5e15 < l

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -0.5 < l < 1.5e15

   1. Initial program 90.0%

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

   2. Taylor expanded in l around 0 90.6%

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

      1. associate-/l* 90.5%

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

      2. associate-/r/ 90.6%

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

      3. unpow2 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1.5e+15)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / (F / ((double) M_PI))) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / (F / Math.PI)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1.5e+15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / (F / math.pi)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1.5e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / Float64(F / pi)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 1.5e+15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / (F / pi)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1.5e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.5e15 < l

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -0.5 < l < 1.5e15

   1. Initial program 90.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in l around 0 98.8%

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

      1. associate-/l* 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 1.5e+15)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 1.5e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 1.5e+15):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 1.5e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * 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 <= 1.5e+15)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 1.5e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 1.5e15 < l

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -0.5 < l < 1.5e15

   1. Initial program 90.0%

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

      1. sqr-neg 90.0%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

      4. sqr-neg 90.7%

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

   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 89.8%

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

      1. unpow2 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in l around 0 89.8%

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

      1. *-lft-identity 89.8%

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

      2. *-lft-identity 89.8%

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

      3. unpow2 89.8%

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

      4. times-frac 89.8%

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

      5. *-lft-identity 89.8%

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

      6. associate-*l/ 89.7%

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

      7. associate-*l* 89.7%

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

      8. unpow-1 89.7%

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

      9. unpow-1 89.7%

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

      10. pow-sqr 89.9%

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

      11. metadata-eval 89.9%

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

      12. distribute-rgt-out-- 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 6: 74.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-108} \lor \neg \left(F \leq 1.16 \cdot 10^{-82}\right) \land \left(F \leq 4.7 \cdot 10^{-36} \lor \neg \left(F \leq 3.6 \cdot 10^{-17}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 1.65e-108)
         (and (not (<= F 1.16e-82)) (or (<= F 4.7e-36) (not (<= F 3.6e-17)))))
   (* PI l)
   (* (/ PI F) (- (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 1.65e-108) || (!(F <= 1.16e-82) && ((F <= 4.7e-36) || !(F <= 3.6e-17)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) / F) * -(l / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 1.65e-108) || (!(F <= 1.16e-82) && ((F <= 4.7e-36) || !(F <= 3.6e-17)))) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI / F) * -(l / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 1.65e-108) or (not (F <= 1.16e-82) and ((F <= 4.7e-36) or not (F <= 3.6e-17))):
		tmp = math.pi * l
	else:
		tmp = (math.pi / F) * -(l / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 1.65e-108) || (!(F <= 1.16e-82) && ((F <= 4.7e-36) || !(F <= 3.6e-17))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi / F) * Float64(-Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 1.65e-108) || (~((F <= 1.16e-82)) && ((F <= 4.7e-36) || ~((F <= 3.6e-17)))))
		tmp = pi * l;
	else
		tmp = (pi / F) * -(l / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 1.65e-108], And[N[Not[LessEqual[F, 1.16e-82]], $MachinePrecision], Or[LessEqual[F, 4.7e-36], N[Not[LessEqual[F, 3.6e-17]], $MachinePrecision]]]], N[(Pi * l), $MachinePrecision], N[(N[(Pi / F), $MachinePrecision] * (-N[(l / F), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.6500000000000001e-108 or 1.16e-82 < F < 4.7000000000000003e-36 or 3.59999999999999995e-17 < F

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. associate-*l/ 74.7%

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

      3. *-lft-identity 74.7%

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

      4. sqr-neg 74.7%

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

   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 66.2%

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

      1. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in F around inf 78.7%

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

   if 1.6500000000000001e-108 < F < 1.16e-82 or 4.7000000000000003e-36 < F < 3.59999999999999995e-17

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in F around 0 79.6%

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

      1. mul-1-neg 79.6%

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

      2. associate-/l* 79.7%

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

      3. unpow2 79.7%

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

      4. associate-/l* 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in l around 0 79.6%

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

       1. unpow2 79.6%

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

       2. times-frac 79.7%

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

       3. *-commutative 79.7%

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

   12. Simplified 79.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-108} \lor \neg \left(F \leq 1.16 \cdot 10^{-82}\right) \land \left(F \leq 4.7 \cdot 10^{-36} \lor \neg \left(F \leq 3.6 \cdot 10^{-17}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \end{array} \]

Alternative 7: 74.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-37} \lor \neg \left(F \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 6.8e-109)
   (* PI l)
   (if (<= F 9e-83)
     (* (/ PI F) (- (/ l F)))
     (if (or (<= F 2.4e-37) (not (<= F 5.5e-17)))
       (* PI l)
       (/ (* PI (- l)) (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 6.8e-109) {
		tmp = ((double) M_PI) * l;
	} else if (F <= 9e-83) {
		tmp = (((double) M_PI) / F) * -(l / F);
	} else if ((F <= 2.4e-37) || !(F <= 5.5e-17)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * -l) / (F * F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 6.8e-109) {
		tmp = Math.PI * l;
	} else if (F <= 9e-83) {
		tmp = (Math.PI / F) * -(l / F);
	} else if ((F <= 2.4e-37) || !(F <= 5.5e-17)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * -l) / (F * F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 6.8e-109:
		tmp = math.pi * l
	elif F <= 9e-83:
		tmp = (math.pi / F) * -(l / F)
	elif (F <= 2.4e-37) or not (F <= 5.5e-17):
		tmp = math.pi * l
	else:
		tmp = (math.pi * -l) / (F * F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 6.8e-109)
		tmp = Float64(pi * l);
	elseif (F <= 9e-83)
		tmp = Float64(Float64(pi / F) * Float64(-Float64(l / F)));
	elseif ((F <= 2.4e-37) || !(F <= 5.5e-17))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * Float64(-l)) / Float64(F * F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 6.8e-109)
		tmp = pi * l;
	elseif (F <= 9e-83)
		tmp = (pi / F) * -(l / F);
	elseif ((F <= 2.4e-37) || ~((F <= 5.5e-17)))
		tmp = pi * l;
	else
		tmp = (pi * -l) / (F * F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 6.8e-109], N[(Pi * l), $MachinePrecision], If[LessEqual[F, 9e-83], N[(N[(Pi / F), $MachinePrecision] * (-N[(l / F), $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 2.4e-37], N[Not[LessEqual[F, 5.5e-17]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * (-l)), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 6.80000000000000023e-109 or 8.99999999999999995e-83 < F < 2.39999999999999991e-37 or 5.50000000000000001e-17 < F

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. associate-*l/ 74.7%

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

      3. *-lft-identity 74.7%

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

      4. sqr-neg 74.7%

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

   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 66.2%

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

      1. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in F around inf 78.7%

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

   if 6.80000000000000023e-109 < F < 8.99999999999999995e-83

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in F around 0 79.4%

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

      1. mul-1-neg 79.4%

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

      2. associate-/l* 80.0%

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

      3. unpow2 80.0%

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

      4. associate-/l* 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in l around 0 79.4%

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

       1. unpow2 79.4%

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

       2. times-frac 80.0%

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

       3. *-commutative 80.0%

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

   12. Simplified 80.0%

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

   if 2.39999999999999991e-37 < F < 5.50000000000000001e-17

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in F around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. neg-mul-1 79.7%

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

      4. *-commutative 79.7%

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

      5. distribute-rgt-neg-in 79.7%

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

      6. unpow2 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-37} \lor \neg \left(F \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 8: 74.1% 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 75.3%

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

   1. sqr-neg 75.3%

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

   2. associate-*l/ 75.7%

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

   3. *-lft-identity 75.7%

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

   4. sqr-neg 75.7%

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

3. Simplified 75.7%

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

4. Taylor expanded in l around 0 66.7%

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

   1. unpow2 66.7%

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

6. Simplified 66.7%

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

7. Taylor expanded in F around inf 76.5%

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

8. Final simplification 76.5%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))