VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 98.9%

Time: 12.0s

Alternatives: 9

Speedup: 0.8×

• could not determine a ground truth

  1. F = -2.924485433011906e-59
  2. l = 2155423413003933.5

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: 76.4% 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: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+25) (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) <= -5e+25) || !((((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) <= -5e+25) || !((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) <= -5e+25) 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) <= -5e+25) || !(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) <= -5e+25) || ~(((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], -5e+25], 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) < -5.00000000000000024e25 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 65.4%

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

   2. Taylor expanded in l around 0 50.4%

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

      1. unpow2 50.4%

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

      2. times-frac 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5.00000000000000024e25 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 85.6%

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

      1. associate-*l/ 87.0%

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

      2. *-un-lft-identity 87.0%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+25) (not (<= (* PI l) 10000000000000.0)))
   (* PI l)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+25) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		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 (((Math.PI * l) <= -5e+25) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+25) or not ((math.pi * l) <= 10000000000000.0):
		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 ((Float64(pi * l) <= -5e+25) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / F) * Float64(pi / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+25) || ~(((pi * l) <= 10000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) * (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+25], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5.00000000000000024e25 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 65.4%

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

   2. Taylor expanded in l around 0 50.4%

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

      1. unpow2 50.4%

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

      2. times-frac 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5.00000000000000024e25 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 85.6%

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

   2. Taylor expanded in l around 0 86.4%

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

      1. unpow2 86.4%

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

      2. times-frac 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+25) (not (<= (* PI l) 10000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+25) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		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 (((Math.PI * l) <= -5e+25) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / (F / Math.PI)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+25) or not ((math.pi * l) <= 10000000000000.0):
		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 ((Float64(pi * l) <= -5e+25) || !(Float64(pi * l) <= 10000000000000.0))
		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 (((pi * l) <= -5e+25) || ~(((pi * l) <= 10000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / (F / pi)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+25], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $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 (*.f64 (PI.f64) l) < -5.00000000000000024e25 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 65.4%

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

   2. Taylor expanded in l around 0 50.4%

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

      1. unpow2 50.4%

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

      2. times-frac 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5.00000000000000024e25 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 85.6%

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

      1. associate-*l/ 87.0%

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

      2. *-un-lft-identity 87.0%

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

      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 99.1%

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

      1. associate-/l* 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 4: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+25) (not (<= (* PI l) 10000000000000.0)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+25) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		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 (((Math.PI * l) <= -5e+25) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+25) or not ((math.pi * l) <= 10000000000000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+25) || !(Float64(pi * l) <= 10000000000000.0))
		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 (((pi * l) <= -5e+25) || ~(((pi * l) <= 10000000000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+25], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $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 (*.f64 (PI.f64) l) < -5.00000000000000024e25 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 65.4%

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

   2. Taylor expanded in l around 0 50.4%

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

      1. unpow2 50.4%

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

      2. times-frac 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5.00000000000000024e25 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 85.6%

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

   2. Taylor expanded in l around 0 86.4%

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

      1. unpow2 86.4%

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

      2. times-frac 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in l around 0 85.1%

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

      1. unpow2 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 5: 50.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{\pi \cdot \frac{-\ell}{F}}{F}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-87} \lor \neg \left(F \leq 1.05 \cdot 10^{-37}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 2.3e-164)
   (/ (* PI (/ (- l) F)) F)
   (if (or (<= F 2.3e-87) (not (<= F 1.05e-37)))
     (* PI l)
     (* l (/ (- PI) (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 2.3e-164) {
		tmp = (((double) M_PI) * (-l / F)) / F;
	} else if ((F <= 2.3e-87) || !(F <= 1.05e-37)) {
		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 <= 2.3e-164) {
		tmp = (Math.PI * (-l / F)) / F;
	} else if ((F <= 2.3e-87) || !(F <= 1.05e-37)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 2.3e-164:
		tmp = (math.pi * (-l / F)) / F
	elif (F <= 2.3e-87) or not (F <= 1.05e-37):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 2.3e-164)
		tmp = Float64(Float64(pi * Float64(Float64(-l) / F)) / F);
	elseif ((F <= 2.3e-87) || !(F <= 1.05e-37))
		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 <= 2.3e-164)
		tmp = (pi * (-l / F)) / F;
	elseif ((F <= 2.3e-87) || ~((F <= 1.05e-37)))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 2.3e-164], N[(N[(Pi * N[((-l) / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision], If[Or[LessEqual[F, 2.3e-87], N[Not[LessEqual[F, 1.05e-37]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.29999999999999985e-164

   1. Initial program 66.7%

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

      1. associate-*l/ 67.8%

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

      2. *-un-lft-identity 67.8%

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

      3. associate-/r* 79.0%

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

   3. Applied egg-rr 79.0%

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

   4. Taylor expanded in l around 0 74.0%

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

   5. Taylor expanded in F around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. unpow2 26.0%

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

      3. associate-*l/ 26.0%

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

      4. *-commutative 26.0%

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

   7. Simplified 26.0%

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

      1. associate-/r* 37.3%

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

      2. frac-2neg 37.3%

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

      3. un-div-inv 37.2%

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

      4. clear-num 37.2%

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

      5. div-inv 37.3%

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

      6. associate-/l* 37.2%

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

      7. un-div-inv 37.3%

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

      8. frac-2neg 37.3%

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

   9. Applied egg-rr 37.3%

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

   if 2.29999999999999985e-164 < F < 2.3000000000000001e-87 or 1.05e-37 < F

   1. Initial program 89.2%

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

   2. Taylor expanded in l around 0 81.5%

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

      1. unpow2 81.5%

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

      2. times-frac 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in F around inf 88.5%

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

   if 2.3000000000000001e-87 < F < 1.05e-37

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in l around 0 69.6%

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

   5. Taylor expanded in F around 0 69.9%

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

      1. mul-1-neg 69.9%

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

      2. unpow2 69.9%

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

      3. associate-*l/ 69.8%

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

      4. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in l around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-*r/ 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{\pi \cdot \frac{-\ell}{F}}{F}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-87} \lor \neg \left(F \leq 1.05 \cdot 10^{-37}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 6: 50.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-85} \lor \neg \left(F \leq 7 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.05e-164)
   (* (/ l F) (/ PI (- F)))
   (if (or (<= F 1.1e-85) (not (<= F 7e-38)))
     (* PI l)
     (* l (/ (- PI) (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.05e-164) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if ((F <= 1.1e-85) || !(F <= 7e-38)) {
		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 <= 1.05e-164) {
		tmp = (l / F) * (Math.PI / -F);
	} else if ((F <= 1.1e-85) || !(F <= 7e-38)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.05e-164:
		tmp = (l / F) * (math.pi / -F)
	elif (F <= 1.1e-85) or not (F <= 7e-38):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.05e-164)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif ((F <= 1.1e-85) || !(F <= 7e-38))
		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 <= 1.05e-164)
		tmp = (l / F) * (pi / -F);
	elseif ((F <= 1.1e-85) || ~((F <= 7e-38)))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.05e-164], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.1e-85], N[Not[LessEqual[F, 7e-38]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.04999999999999995e-164

   1. Initial program 66.7%

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

   2. Taylor expanded in l around 0 62.7%

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

      1. unpow2 62.7%

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

      2. times-frac 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in F around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. *-commutative 26.0%

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

      3. unpow2 26.0%

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

      4. times-frac 37.3%

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

      5. distribute-lft-neg-out 37.3%

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

      6. neg-mul-1 37.3%

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

      7. metadata-eval 37.3%

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

      8. times-frac 37.3%

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

      9. *-lft-identity 37.3%

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

      10. neg-mul-1 37.3%

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

   7. Simplified 37.3%

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

   if 1.04999999999999995e-164 < F < 1.1e-85 or 7.0000000000000003e-38 < F

   1. Initial program 89.2%

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

   2. Taylor expanded in l around 0 81.5%

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

      1. unpow2 81.5%

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

      2. times-frac 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in F around inf 88.5%

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

   if 1.1e-85 < F < 7.0000000000000003e-38

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in l around 0 69.6%

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

   5. Taylor expanded in F around 0 69.9%

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

      1. mul-1-neg 69.9%

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

      2. unpow2 69.9%

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

      3. associate-*l/ 69.8%

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

      4. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in l around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-*r/ 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-85} \lor \neg \left(F \leq 7 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 7: 50.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-84} \lor \neg \left(F \leq 9.5 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.2e-164)
   (* (/ l F) (/ PI (- F)))
   (if (or (<= F 9.5e-84) (not (<= F 9.5e-38)))
     (* PI l)
     (/ (/ l (/ F PI)) (- F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.2e-164) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if ((F <= 9.5e-84) || !(F <= 9.5e-38)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l / (F / ((double) M_PI))) / -F;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 1.2e-164) {
		tmp = (l / F) * (Math.PI / -F);
	} else if ((F <= 9.5e-84) || !(F <= 9.5e-38)) {
		tmp = Math.PI * l;
	} else {
		tmp = (l / (F / Math.PI)) / -F;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.2e-164:
		tmp = (l / F) * (math.pi / -F)
	elif (F <= 9.5e-84) or not (F <= 9.5e-38):
		tmp = math.pi * l
	else:
		tmp = (l / (F / math.pi)) / -F
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.2e-164)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif ((F <= 9.5e-84) || !(F <= 9.5e-38))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l / Float64(F / pi)) / Float64(-F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.2e-164)
		tmp = (l / F) * (pi / -F);
	elseif ((F <= 9.5e-84) || ~((F <= 9.5e-38)))
		tmp = pi * l;
	else
		tmp = (l / (F / pi)) / -F;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.2e-164], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 9.5e-84], N[Not[LessEqual[F, 9.5e-38]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.19999999999999992e-164

   1. Initial program 66.7%

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

   2. Taylor expanded in l around 0 62.7%

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

      1. unpow2 62.7%

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

      2. times-frac 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in F around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. *-commutative 26.0%

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

      3. unpow2 26.0%

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

      4. times-frac 37.3%

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

      5. distribute-lft-neg-out 37.3%

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

      6. neg-mul-1 37.3%

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

      7. metadata-eval 37.3%

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

      8. times-frac 37.3%

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

      9. *-lft-identity 37.3%

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

      10. neg-mul-1 37.3%

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

   7. Simplified 37.3%

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

   if 1.19999999999999992e-164 < F < 9.49999999999999941e-84 or 9.5000000000000009e-38 < F

   1. Initial program 89.2%

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

   2. Taylor expanded in l around 0 81.5%

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

      1. unpow2 81.5%

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

      2. times-frac 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in F around inf 88.5%

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

   if 9.49999999999999941e-84 < F < 9.5000000000000009e-38

   1. Initial program 99.7%

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

   2. Taylor expanded in l around 0 69.9%

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

      1. unpow2 69.9%

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

      2. times-frac 69.8%

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

   4. Simplified 69.8%

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

   5. Taylor expanded in F around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-/l/ 69.6%

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

      3. frac-2neg 69.6%

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

      4. distribute-frac-neg 69.6%

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

      5. associate-*l/ 69.8%

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

      6. *-commutative 69.8%

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

   7. Applied egg-rr 69.8%

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

      1. frac-2neg 69.8%

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

      2. un-div-inv 69.8%

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

      3. *-commutative 69.8%

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

      4. un-div-inv 69.8%

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

      5. frac-2neg 69.8%

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

      6. associate-/r/ 69.6%

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

   9. Applied egg-rr 69.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-84} \lor \neg \left(F \leq 9.5 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \end{array} \]

Alternative 8: 74.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.3 \cdot 10^{-82} \lor \neg \left(F \leq 5 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 3.3e-82) (not (<= F 5e-38))) (* PI l) (* l (/ (- PI) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 3.3e-82) || !(F <= 5e-38)) {
		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 <= 3.3e-82) || !(F <= 5e-38)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 3.3e-82) or not (F <= 5e-38):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 3.3e-82) || !(F <= 5e-38))
		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 <= 3.3e-82) || ~((F <= 5e-38)))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 3.3e-82], N[Not[LessEqual[F, 5e-38]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.30000000000000022e-82 or 5.00000000000000033e-38 < F

   1. Initial program 75.4%

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

   2. Taylor expanded in l around 0 70.0%

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

      1. unpow2 70.0%

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

      2. times-frac 77.1%

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

   4. Simplified 77.1%

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

   5. Taylor expanded in F around inf 73.4%

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

   if 3.30000000000000022e-82 < F < 5.00000000000000033e-38

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in l around 0 69.6%

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

   5. Taylor expanded in F around 0 69.9%

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

      1. mul-1-neg 69.9%

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

      2. unpow2 69.9%

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

      3. associate-*l/ 69.8%

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

      4. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in l around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-*r/ 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.3 \cdot 10^{-82} \lor \neg \left(F \leq 5 \cdot 10^{-38}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 9: 73.6% 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 76.3%

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

2. Taylor expanded in l around 0 70.0%

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

   1. unpow2 70.0%

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

   2. times-frac 76.8%

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

4. Simplified 76.8%

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

5. Taylor expanded in F around inf 71.8%

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

6. Final simplification 71.8%

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

Reproduce

herbie shell --seed 2023258 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))