VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 98.3%

Time: 13.1s

Alternatives: 11

Speedup: 2.7×

• could not determine a ground truth

  1. F = 2.4568969078966367e+167
  2. l = -5103891088590.5

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: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+32} \lor \neg \left(\pi \cdot \ell \leq 100\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -4e+32) (not (<= (* PI l) 100.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -4e+32) || !((((double) M_PI) * l) <= 100.0)) {
		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) <= -4e+32) || !((Math.PI * l) <= 100.0)) {
		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) <= -4e+32) or not ((math.pi * l) <= 100.0):
		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) <= -4e+32) || !(Float64(pi * l) <= 100.0))
		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) <= -4e+32) || ~(((pi * l) <= 100.0)))
		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], -4e+32], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 100.0]], $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) < -4.00000000000000021e32 or 100 < (*.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. associate-*l/ 63.2%

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

      2. *-lft-identity 63.2%

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

   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 52.0%

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

      1. unpow2 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -4.00000000000000021e32 < (*.f64 (PI.f64) l) < 100

   1. Initial program 88.7%

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

      1. associate-*l/ 89.2%

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

      2. *-un-lft-identity 89.2%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+32} \lor \neg \left(\pi \cdot \ell \leq 100\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+32} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -4e+32) (not (<= (* PI l) 1e-16)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -4e+32) || !((((double) M_PI) * l) <= 1e-16)) {
		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) <= -4e+32) || !((Math.PI * l) <= 1e-16)) {
		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) <= -4e+32) or not ((math.pi * l) <= 1e-16):
		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) <= -4e+32) || !(Float64(pi * l) <= 1e-16))
		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) <= -4e+32) || ~(((pi * l) <= 1e-16)))
		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], -4e+32], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-16]], $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) < -4.00000000000000021e32 or 9.9999999999999998e-17 < (*.f64 (PI.f64) l)

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -4.00000000000000021e32 < (*.f64 (PI.f64) l) < 9.9999999999999998e-17

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-un-lft-identity 89.0%

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

      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 98.8%

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

      1. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+32} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 3: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi + \frac{\pi}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (* l (+ PI (* (/ PI F) (/ -1.0 F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) + ((((double) M_PI) / F) * (-1.0 / F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI + ((Math.PI / F) * (-1.0 / F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi + ((math.pi / F) * (-1.0 / F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi + Float64(Float64(pi / F) * Float64(-1.0 / F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = l * (pi + ((pi / F) * (-1.0 / F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi + N[(N[(Pi / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-lft-identity 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

      1. associate-/r* 87.8%

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

      2. div-inv 87.8%

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

   8. Applied egg-rr 87.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi + \frac{\pi}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \]

Alternative 4: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -460000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -460000000000.0) (not (<= l 0.017)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -460000000000.0) || !(l <= 0.017)) {
		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 <= -460000000000.0) || !(l <= 0.017)) {
		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 <= -460000000000.0) or not (l <= 0.017):
		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 <= -460000000000.0) || !(l <= 0.017))
		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 <= -460000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -460000000000.0], N[Not[LessEqual[l, 0.017]], $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 < -4.6e11 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -4.6e11 < l < 0.017000000000000001

   1. Initial program 88.5%

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

   2. Taylor expanded in l around 0 88.2%

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

      1. associate-/l* 88.1%

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

      2. associate-/r/ 88.2%

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

      3. unpow2 88.2%

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

   4. Simplified 88.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -460000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-lft-identity 89.0%

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

   3. Simplified 89.0%

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

      1. add-cbrt-cube 59.4%

         \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(\sqrt[3]{\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \left(\pi \cdot \ell\right)}\right)}}{F \cdot F} \]

      2. pow1/3 38.3%

         \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left({\left(\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \left(\pi \cdot \ell\right)\right)}^{0.3333333333333333}\right)}}{F \cdot F} \]

      3. pow3 38.3%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left({\color{blue}{\left({\left(\pi \cdot \ell\right)}^{3}\right)}}^{0.3333333333333333}\right)}{F \cdot F} \]

      4. pow-to-exp 25.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left({\color{blue}{\left(e^{\log \left(\pi \cdot \ell\right) \cdot 3}\right)}}^{0.3333333333333333}\right)}{F \cdot F} \]

      5. pow-exp 38.0%

         \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(e^{\left(\log \left(\pi \cdot \ell\right) \cdot 3\right) \cdot 0.3333333333333333}\right)}}{F \cdot F} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(e^{\left(\log \left(\pi \cdot \ell\right) \cdot 3\right) \cdot 0.3333333333333333}\right)}}{F \cdot F} \]

   6. Taylor expanded in l around 0 88.2%

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

      1. *-commutative 88.2%

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

      2. unpow2 88.2%

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

      3. times-frac 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (- (* PI l) (/ (* l (/ PI F)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l * (((double) M_PI) / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l * (Math.PI / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l * (math.pi / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l * Float64(pi / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l * (pi / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-un-lft-identity 89.0%

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

      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 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. times-frac 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in l around 0 98.8%

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

      1. associate-*r/ 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \]

Alternative 7: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-lft-identity 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 8: 74.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-18} \lor \neg \left(\ell \leq -4.3 \cdot 10^{-216} \lor \neg \left(\ell \leq -2.45 \cdot 10^{-238}\right) \land \ell \leq 8.5 \cdot 10^{-13}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -4e-18)
         (not
          (or (<= l -4.3e-216) (and (not (<= l -2.45e-238)) (<= l 8.5e-13)))))
   (* PI l)
   (* (/ PI F) (/ (- l) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -4e-18) || !((l <= -4.3e-216) || (!(l <= -2.45e-238) && (l <= 8.5e-13)))) {
		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 ((l <= -4e-18) || !((l <= -4.3e-216) || (!(l <= -2.45e-238) && (l <= 8.5e-13)))) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI / F) * (-l / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -4e-18) or not ((l <= -4.3e-216) or (not (l <= -2.45e-238) and (l <= 8.5e-13))):
		tmp = math.pi * l
	else:
		tmp = (math.pi / F) * (-l / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -4e-18) || !((l <= -4.3e-216) || (!(l <= -2.45e-238) && (l <= 8.5e-13))))
		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 ((l <= -4e-18) || ~(((l <= -4.3e-216) || (~((l <= -2.45e-238)) && (l <= 8.5e-13)))))
		tmp = pi * l;
	else
		tmp = (pi / F) * (-l / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -4e-18], N[Not[Or[LessEqual[l, -4.3e-216], And[N[Not[LessEqual[l, -2.45e-238]], $MachinePrecision], LessEqual[l, 8.5e-13]]]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.0000000000000003e-18 or -4.2999999999999997e-216 < l < -2.4499999999999999e-238 or 8.5000000000000001e-13 < l

   1. Initial program 68.0%

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

      1. associate-*l/ 68.0%

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

      2. *-lft-identity 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in l around 0 57.1%

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

      1. unpow2 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in F around inf 96.8%

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

   if -4.0000000000000003e-18 < l < -4.2999999999999997e-216 or -2.4499999999999999e-238 < l < 8.5000000000000001e-13

   1. Initial program 87.5%

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

      1. associate-*l/ 88.1%

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

      2. *-lft-identity 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in l around 0 87.6%

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

      1. unpow2 87.6%

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

   6. Simplified 87.6%

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

      1. sub-neg 87.6%

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

      2. distribute-lft-in 87.5%

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

      3. *-commutative 87.5%

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

      4. div-inv 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. pow2 87.5%

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

      7. pow-flip 87.5%

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

      8. metadata-eval 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. *-commutative 87.5%

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

      2. distribute-lft-in 87.5%

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

      3. distribute-rgt-neg-out 87.5%

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

      4. sub-neg 87.5%

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

      5. *-rgt-identity 87.5%

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

      6. distribute-lft-out-- 87.5%

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

   10. Simplified 87.5%

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

   11. Taylor expanded in F around 0 54.3%

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

       1. mul-1-neg 54.3%

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

       2. associate-/l* 54.2%

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

       3. unpow2 54.2%

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

   13. Simplified 54.2%

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

       1. associate-/l* 54.1%

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

       2. associate-/r/ 65.7%

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

   15. Applied egg-rr 65.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-18} \lor \neg \left(\ell \leq -4.3 \cdot 10^{-216} \lor \neg \left(\ell \leq -2.45 \cdot 10^{-238}\right) \land \ell \leq 8.5 \cdot 10^{-13}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \]

Alternative 9: 92.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + \frac{-1}{F \cdot F}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (* l (* PI (+ 1.0 (/ -1.0 (* F F)))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 + (-1.0 / (F * F))));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 + (-1.0 / (F * F))));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 + (-1.0 / (F * F))))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * Float64(1.0 + Float64(-1.0 / Float64(F * F)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 + (-1.0 / (F * F))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi * N[(1.0 + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-lft-identity 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

      1. sub-neg 87.8%

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

      2. distribute-lft-in 87.7%

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

      3. *-commutative 87.7%

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

      4. div-inv 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

      6. pow2 87.7%

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

      7. pow-flip 87.7%

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

      8. metadata-eval 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

      2. distribute-lft-in 87.7%

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

      3. distribute-rgt-neg-out 87.7%

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

      4. sub-neg 87.7%

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

      5. *-rgt-identity 87.7%

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

      6. distribute-lft-out-- 87.7%

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

   10. Simplified 87.7%

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

   11. Taylor expanded in l around 0 87.7%

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

       1. *-commutative 87.7%

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

       2. /-rgt-identity 87.7%

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

       3. unpow2 87.7%

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

       4. associate-/l* 87.7%

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

       5. associate-/r/ 87.6%

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

       6. unpow-1 87.6%

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

       7. unpow-1 87.6%

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

       8. pow-sqr 87.7%

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

       9. metadata-eval 87.7%

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

       10. metadata-eval 87.7%

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

       11. pow-sqr 87.6%

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

       12. unpow-1 87.6%

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

       13. unpow-1 87.6%

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

       14. associate-/r/ 87.7%

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

       15. associate-/l* 87.7%

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

       16. unpow2 87.7%

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

       17. /-rgt-identity 87.7%

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

       18. unpow2 87.7%

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

   13. Simplified 87.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + \frac{-1}{F \cdot F}\right)\right)\\ \end{array} \]

Alternative 10: 92.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 + \frac{-1}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7200000000000.0) (not (<= l 0.017)))
   (* PI l)
   (* (* PI l) (+ 1.0 (/ -1.0 (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) * (1.0 + (-1.0 / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7200000000000.0) || !(l <= 0.017)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) * (1.0 + (-1.0 / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7200000000000.0) or not (l <= 0.017):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) * (1.0 + (-1.0 / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7200000000000.0) || !(l <= 0.017))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) * Float64(1.0 + Float64(-1.0 / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7200000000000.0) || ~((l <= 0.017)))
		tmp = pi * l;
	else
		tmp = (pi * l) * (1.0 + (-1.0 / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7200000000000.0], N[Not[LessEqual[l, 0.017]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] * N[(1.0 + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e12 or 0.017000000000000001 < l

   1. Initial program 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -7.2e12 < l < 0.017000000000000001

   1. Initial program 88.5%

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

      1. associate-*l/ 89.0%

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

      2. *-lft-identity 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

      1. sub-neg 87.8%

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

      2. distribute-lft-in 87.7%

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

      3. *-commutative 87.7%

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

      4. div-inv 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

      6. pow2 87.7%

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

      7. pow-flip 87.7%

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

      8. metadata-eval 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

      2. distribute-lft-in 87.7%

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

      3. distribute-rgt-neg-out 87.7%

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

      4. sub-neg 87.7%

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

      5. *-rgt-identity 87.7%

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

      6. distribute-lft-out-- 87.7%

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

   10. Simplified 87.7%

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

   11. Taylor expanded in l around 0 87.7%

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

       1. *-commutative 87.7%

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

       2. /-rgt-identity 87.7%

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

       3. unpow2 87.7%

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

       4. associate-/l* 87.7%

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

       5. associate-/r/ 87.6%

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

       6. unpow-1 87.6%

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

       7. unpow-1 87.6%

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

       8. pow-sqr 87.7%

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

       9. metadata-eval 87.7%

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

       10. metadata-eval 87.7%

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

       11. pow-sqr 87.6%

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

       12. unpow-1 87.6%

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

       13. unpow-1 87.6%

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

       14. associate-/r/ 87.7%

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

       15. associate-/l* 87.7%

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

       16. unpow2 87.7%

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

       17. /-rgt-identity 87.7%

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

       18. unpow2 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in l around 0 87.7%

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

       1. *-commutative 87.7%

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

       2. associate-*l* 87.8%

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

       3. sub-neg 87.8%

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

       4. distribute-neg-frac 87.8%

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

       5. metadata-eval 87.8%

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

       6. unpow2 87.8%

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

       7. *-commutative 87.8%

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

   16. Simplified 87.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000000000 \lor \neg \left(\ell \leq 0.017\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 + \frac{-1}{F \cdot F}\right)\\ \end{array} \]

Alternative 11: 73.4% 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 76.9%

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

   1. associate-*l/ 77.1%

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

   2. *-lft-identity 77.1%

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

3. Simplified 77.1%

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

4. Taylor expanded in l around 0 70.9%

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

   1. unpow2 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in F around inf 68.3%

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

8. Final simplification 68.3%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))