VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.2% → 99.0%

Time: 23.8s

Alternatives: 11

Speedup: 2.7×

• could not determine a ground truth

  1. F = -1.8699987208296056e-5
  2. l = 3023228593631944.5

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 200000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+26) (not (<= (* PI l) 200000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+26) || !((((double) M_PI) * l) <= 200000000000.0)) {
		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+26) || !((Math.PI * l) <= 200000000000.0)) {
		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+26) or not ((math.pi * l) <= 200000000000.0):
		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+26) || !(Float64(pi * l) <= 200000000000.0))
		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+26) || ~(((pi * l) <= 200000000000.0)))
		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+26], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 200000000000.0]], $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) < -5.0000000000000001e26 or 2e11 < (*.f64 (PI.f64) l)

   1. Initial program 56.9%

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

      1. sqr-neg 56.9%

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

      2. associate-*l/ 56.9%

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

      3. *-lft-identity 56.9%

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

      4. sqr-neg 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in l around 0 44.2%

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

      1. unpow2 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5.0000000000000001e26 < (*.f64 (PI.f64) l) < 2e11

   1. Initial program 86.7%

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

      1. associate-*l/ 87.7%

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

      2. *-un-lft-identity 87.7%

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

      3. associate-/r* 99.2%

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

   3. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 200000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+26) (not (<= (* PI l) 5e-16)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+26) || !((((double) M_PI) * l) <= 5e-16)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((((double) M_PI) * l) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+26) || !((Math.PI * l) <= 5e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (((Math.PI * l) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+26) or not ((math.pi * l) <= 5e-16):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (((math.pi * l) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+26) || !(Float64(pi * l) <= 5e-16))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+26) || ~(((pi * l) <= 5e-16)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (((pi * l) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+26], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5.0000000000000001e26 or 5.0000000000000004e-16 < (*.f64 (PI.f64) l)

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -5.0000000000000001e26 < (*.f64 (PI.f64) l) < 5.0000000000000004e-16

   1. Initial program 86.4%

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

      1. associate-*l/ 87.6%

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

      2. *-un-lft-identity 87.6%

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

      3. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in l around 0 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 3: 92.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.8e+22) (not (<= l 1.25e-15)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1.8e+22) or not (l <= 1.25e-15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1.8e+22) || !(l <= 1.25e-15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1.8e+22) || ~((l <= 1.25e-15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.8e+22], N[Not[LessEqual[l, 1.25e-15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8e22 or 1.25e-15 < l

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -1.8e22 < l < 1.25e-15

   1. Initial program 86.4%

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

   2. Taylor expanded in l around 0 87.6%

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

      1. associate-/l* 87.4%

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

      2. associate-/r/ 87.6%

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

      3. unpow2 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.8e+22) (not (<= l 1.25e-15)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		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 <= -1.8e+22) || !(l <= 1.25e-15)) {
		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 <= -1.8e+22) or not (l <= 1.25e-15):
		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 <= -1.8e+22) || !(l <= 1.25e-15))
		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 <= -1.8e+22) || ~((l <= 1.25e-15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.8e+22], N[Not[LessEqual[l, 1.25e-15]], $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 < -1.8e22 or 1.25e-15 < l

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -1.8e22 < l < 1.25e-15

   1. Initial program 86.4%

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

      1. associate-*l/ 87.6%

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

      2. *-un-lft-identity 87.6%

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

      3. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in l around 0 87.6%

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

      1. *-commutative 87.6%

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

      2. unpow2 87.6%

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

      3. times-frac 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 5: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.8e+22) (not (<= l 1.25e-15)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1.8e+22) or not (l <= 1.25e-15):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1.8e+22) || !(l <= 1.25e-15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1.8e+22) || ~((l <= 1.25e-15)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.8e+22], N[Not[LessEqual[l, 1.25e-15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8e22 or 1.25e-15 < l

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -1.8e22 < l < 1.25e-15

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. associate-*l/ 87.6%

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

      3. *-lft-identity 87.6%

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

      4. sqr-neg 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in l around 0 86.5%

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

      1. unpow2 86.5%

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

   6. Simplified 86.5%

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

      1. sub-neg 86.5%

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

      2. distribute-lft-in 86.5%

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

      3. *-commutative 86.5%

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

      4. div-inv 86.5%

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

      5. distribute-rgt-neg-in 86.5%

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

      6. pow2 86.5%

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

      7. pow-flip 86.5%

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

      8. metadata-eval 86.5%

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

   8. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. distribute-rgt-out 86.5%

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

      3. distribute-rgt-neg-out 86.5%

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

      4. sub-neg 86.5%

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

      5. *-rgt-identity 86.5%

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

      6. distribute-lft-out-- 86.5%

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

   10. Simplified 86.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 6: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.8e+22) (not (<= l 1.25e-15)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		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 <= -1.8e+22) || !(l <= 1.25e-15)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1.8e+22) or not (l <= 1.25e-15):
		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 <= -1.8e+22) || !(l <= 1.25e-15))
		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 <= -1.8e+22) || ~((l <= 1.25e-15)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.8e+22], N[Not[LessEqual[l, 1.25e-15]], $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 < -1.8e22 or 1.25e-15 < l

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -1.8e22 < l < 1.25e-15

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. associate-*l/ 87.6%

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

      3. *-lft-identity 87.6%

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

      4. sqr-neg 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in l around 0 86.5%

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

      1. unpow2 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{if}\;\ell \leq -118000:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-203}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{-183}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot \frac{F}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-116} \lor \neg \left(\ell \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* l (/ (- PI) (* F F)))))
   (if (<= l -118000.0)
     (* PI l)
     (if (<= l -7e-92)
       t_0
       (if (<= l -2.4e-203)
         (* PI l)
         (if (<= l 2.95e-183)
           (/ PI (* (- F) (/ F l)))
           (if (or (<= l 2.9e-116) (not (<= l 3.4e-16))) (* PI l) t_0)))))))

C

double code(double F, double l) {
	double t_0 = l * (-((double) M_PI) / (F * F));
	double tmp;
	if (l <= -118000.0) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -7e-92) {
		tmp = t_0;
	} else if (l <= -2.4e-203) {
		tmp = ((double) M_PI) * l;
	} else if (l <= 2.95e-183) {
		tmp = ((double) M_PI) / (-F * (F / l));
	} else if ((l <= 2.9e-116) || !(l <= 3.4e-16)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = l * (-Math.PI / (F * F));
	double tmp;
	if (l <= -118000.0) {
		tmp = Math.PI * l;
	} else if (l <= -7e-92) {
		tmp = t_0;
	} else if (l <= -2.4e-203) {
		tmp = Math.PI * l;
	} else if (l <= 2.95e-183) {
		tmp = Math.PI / (-F * (F / l));
	} else if ((l <= 2.9e-116) || !(l <= 3.4e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = l * (-math.pi / (F * F))
	tmp = 0
	if l <= -118000.0:
		tmp = math.pi * l
	elif l <= -7e-92:
		tmp = t_0
	elif l <= -2.4e-203:
		tmp = math.pi * l
	elif l <= 2.95e-183:
		tmp = math.pi / (-F * (F / l))
	elif (l <= 2.9e-116) or not (l <= 3.4e-16):
		tmp = math.pi * l
	else:
		tmp = t_0
	return tmp

Julia

function code(F, l)
	t_0 = Float64(l * Float64(Float64(-pi) / Float64(F * F)))
	tmp = 0.0
	if (l <= -118000.0)
		tmp = Float64(pi * l);
	elseif (l <= -7e-92)
		tmp = t_0;
	elseif (l <= -2.4e-203)
		tmp = Float64(pi * l);
	elseif (l <= 2.95e-183)
		tmp = Float64(pi / Float64(Float64(-F) * Float64(F / l)));
	elseif ((l <= 2.9e-116) || !(l <= 3.4e-16))
		tmp = Float64(pi * l);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = l * (-pi / (F * F));
	tmp = 0.0;
	if (l <= -118000.0)
		tmp = pi * l;
	elseif (l <= -7e-92)
		tmp = t_0;
	elseif (l <= -2.4e-203)
		tmp = pi * l;
	elseif (l <= 2.95e-183)
		tmp = pi / (-F * (F / l));
	elseif ((l <= 2.9e-116) || ~((l <= 3.4e-16)))
		tmp = pi * l;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -118000.0], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -7e-92], t$95$0, If[LessEqual[l, -2.4e-203], N[(Pi * l), $MachinePrecision], If[LessEqual[l, 2.95e-183], N[(Pi / N[((-F) * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.9e-116], N[Not[LessEqual[l, 3.4e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -118000 or -7e-92 < l < -2.3999999999999999e-203 or 2.94999999999999992e-183 < l < 2.8999999999999998e-116 or 3.4e-16 < l

   1. Initial program 65.8%

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

      1. sqr-neg 65.8%

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

      2. associate-*l/ 66.0%

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

      3. *-lft-identity 66.0%

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

      4. sqr-neg 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in l around 0 55.0%

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

      1. unpow2 55.0%

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

   6. Simplified 55.0%

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

   7. Taylor expanded in F around inf 90.7%

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

   if -118000 < l < -7e-92 or 2.8999999999999998e-116 < l < 3.4e-16

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in F around 0 66.6%

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

      1. associate-*r/ 66.6%

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

      2. neg-mul-1 66.6%

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

      3. unpow2 66.6%

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

   9. Simplified 66.6%

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

   if -2.3999999999999999e-203 < l < 2.94999999999999992e-183

   1. Initial program 73.8%

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

      1. sqr-neg 73.8%

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

      2. associate-*l/ 76.1%

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

      3. *-lft-identity 76.1%

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

      4. sqr-neg 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in l around 0 74.0%

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

      1. unpow2 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in F around 0 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

      3. unpow2 39.4%

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

   9. Simplified 39.4%

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

       1. clear-num 39.4%

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

       2. un-div-inv 41.2%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 3.9%

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

       5. sqr-neg 3.9%

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

       6. sqrt-unprod 3.9%

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

       7. add-sqr-sqrt 3.9%

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

       8. associate-/l* 4.0%

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

       9. frac-times 4.0%

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

       10. clear-num 4.0%

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

       11. frac-2neg 4.0%

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

       12. frac-times 4.0%

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

       13. *-un-lft-identity 4.0%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 65.1%

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

       16. sqr-neg 65.1%

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

       17. sqrt-unprod 64.4%

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

       18. add-sqr-sqrt 65.1%

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

   11. Applied egg-rr 65.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -118000:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-92}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-203}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{-183}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot \frac{F}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-116} \lor \neg \left(\ell \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 8: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -16500:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\pi \cdot \frac{\ell}{F}}{-F}\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-203}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-182}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot \frac{F}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-116} \lor \neg \left(\ell \leq 3.6 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -16500.0)
   (* PI l)
   (if (<= l -5.8e-93)
     (/ (* PI (/ l F)) (- F))
     (if (<= l -4.6e-203)
       (* PI l)
       (if (<= l 5.6e-182)
         (/ PI (* (- F) (/ F l)))
         (if (or (<= l 2.5e-116) (not (<= l 3.6e-16)))
           (* PI l)
           (* l (/ (- PI) (* F F)))))))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -16500.0) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -5.8e-93) {
		tmp = (((double) M_PI) * (l / F)) / -F;
	} else if (l <= -4.6e-203) {
		tmp = ((double) M_PI) * l;
	} else if (l <= 5.6e-182) {
		tmp = ((double) M_PI) / (-F * (F / l));
	} else if ((l <= 2.5e-116) || !(l <= 3.6e-16)) {
		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 (l <= -16500.0) {
		tmp = Math.PI * l;
	} else if (l <= -5.8e-93) {
		tmp = (Math.PI * (l / F)) / -F;
	} else if (l <= -4.6e-203) {
		tmp = Math.PI * l;
	} else if (l <= 5.6e-182) {
		tmp = Math.PI / (-F * (F / l));
	} else if ((l <= 2.5e-116) || !(l <= 3.6e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -16500.0:
		tmp = math.pi * l
	elif l <= -5.8e-93:
		tmp = (math.pi * (l / F)) / -F
	elif l <= -4.6e-203:
		tmp = math.pi * l
	elif l <= 5.6e-182:
		tmp = math.pi / (-F * (F / l))
	elif (l <= 2.5e-116) or not (l <= 3.6e-16):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -16500.0)
		tmp = Float64(pi * l);
	elseif (l <= -5.8e-93)
		tmp = Float64(Float64(pi * Float64(l / F)) / Float64(-F));
	elseif (l <= -4.6e-203)
		tmp = Float64(pi * l);
	elseif (l <= 5.6e-182)
		tmp = Float64(pi / Float64(Float64(-F) * Float64(F / l)));
	elseif ((l <= 2.5e-116) || !(l <= 3.6e-16))
		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 (l <= -16500.0)
		tmp = pi * l;
	elseif (l <= -5.8e-93)
		tmp = (pi * (l / F)) / -F;
	elseif (l <= -4.6e-203)
		tmp = pi * l;
	elseif (l <= 5.6e-182)
		tmp = pi / (-F * (F / l));
	elseif ((l <= 2.5e-116) || ~((l <= 3.6e-16)))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -16500.0], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -5.8e-93], N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision], If[LessEqual[l, -4.6e-203], N[(Pi * l), $MachinePrecision], If[LessEqual[l, 5.6e-182], N[(Pi / N[((-F) * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.5e-116], N[Not[LessEqual[l, 3.6e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -16500 or -5.7999999999999997e-93 < l < -4.59999999999999983e-203 or 5.59999999999999986e-182 < l < 2.5000000000000001e-116 or 3.59999999999999983e-16 < l

   1. Initial program 65.8%

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

      1. sqr-neg 65.8%

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

      2. associate-*l/ 66.0%

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

      3. *-lft-identity 66.0%

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

      4. sqr-neg 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in l around 0 55.0%

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

      1. unpow2 55.0%

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

   6. Simplified 55.0%

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

   7. Taylor expanded in F around inf 90.7%

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

   if -16500 < l < -5.7999999999999997e-93

   1. Initial program 99.3%

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

      1. sqr-neg 99.3%

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

      2. associate-*l/ 99.5%

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

      3. *-lft-identity 99.5%

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

      4. sqr-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in F around 0 62.2%

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

      1. associate-*r/ 62.2%

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

      2. neg-mul-1 62.2%

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

      3. unpow2 62.2%

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

   9. Simplified 62.2%

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

       1. clear-num 62.2%

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

       2. un-div-inv 62.2%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 1.7%

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

       5. sqr-neg 1.7%

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

       6. sqrt-unprod 1.7%

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

       7. add-sqr-sqrt 1.7%

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

       8. associate-/l* 1.7%

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

       9. frac-times 1.7%

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

       10. frac-2neg 1.7%

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

       11. associate-*r/ 1.7%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 62.3%

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

       14. sqr-neg 62.3%

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

       15. sqrt-unprod 62.1%

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

       16. add-sqr-sqrt 62.3%

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

       17. *-commutative 62.3%

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

   11. Applied egg-rr 62.3%

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

   if -4.59999999999999983e-203 < l < 5.59999999999999986e-182

   1. Initial program 73.8%

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

      1. sqr-neg 73.8%

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

      2. associate-*l/ 76.1%

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

      3. *-lft-identity 76.1%

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

      4. sqr-neg 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in l around 0 74.0%

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

      1. unpow2 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in F around 0 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

      3. unpow2 39.4%

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

   9. Simplified 39.4%

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

       1. clear-num 39.4%

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

       2. un-div-inv 41.2%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 3.9%

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

       5. sqr-neg 3.9%

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

       6. sqrt-unprod 3.9%

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

       7. add-sqr-sqrt 3.9%

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

       8. associate-/l* 4.0%

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

       9. frac-times 4.0%

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

       10. clear-num 4.0%

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

       11. frac-2neg 4.0%

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

       12. frac-times 4.0%

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

       13. *-un-lft-identity 4.0%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 65.1%

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

       16. sqr-neg 65.1%

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

       17. sqrt-unprod 64.4%

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

       18. add-sqr-sqrt 65.1%

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

   11. Applied egg-rr 65.1%

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

   if 2.5000000000000001e-116 < l < 3.59999999999999983e-16

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in F around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

      3. unpow2 71.3%

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

   9. Simplified 71.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{-\pi}{F \cdot F}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -16500:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\pi \cdot \frac{\ell}{F}}{-F}\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-203}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-182}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot \frac{F}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-116} \lor \neg \left(\ell \leq 3.6 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 9: 73.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.66 \cdot 10^{-139} \lor \neg \left(F \leq 2.3 \cdot 10^{-127} \lor \neg \left(F \leq 2.55 \cdot 10^{-64}\right) \land F \leq 2.1 \cdot 10^{-42}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 1.66e-139)
         (not (or (<= F 2.3e-127) (and (not (<= F 2.55e-64)) (<= F 2.1e-42)))))
   (* PI l)
   (* l (/ (- PI) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 1.66e-139) || !((F <= 2.3e-127) || (!(F <= 2.55e-64) && (F <= 2.1e-42)))) {
		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 <= 1.66e-139) || !((F <= 2.3e-127) || (!(F <= 2.55e-64) && (F <= 2.1e-42)))) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 1.66e-139) or not ((F <= 2.3e-127) or (not (F <= 2.55e-64) and (F <= 2.1e-42))):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 1.66e-139) || !((F <= 2.3e-127) || (!(F <= 2.55e-64) && (F <= 2.1e-42))))
		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 <= 1.66e-139) || ~(((F <= 2.3e-127) || (~((F <= 2.55e-64)) && (F <= 2.1e-42)))))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 1.66e-139], N[Not[Or[LessEqual[F, 2.3e-127], And[N[Not[LessEqual[F, 2.55e-64]], $MachinePrecision], LessEqual[F, 2.1e-42]]]], $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 < 1.6599999999999999e-139 or 2.30000000000000019e-127 < F < 2.54999999999999992e-64 or 2.10000000000000006e-42 < F

   1. Initial program 71.0%

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

      1. sqr-neg 71.0%

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

      2. associate-*l/ 71.6%

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

      3. *-lft-identity 71.6%

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

      4. sqr-neg 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in l around 0 63.5%

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

      1. unpow2 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in F around inf 75.7%

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

   if 1.6599999999999999e-139 < F < 2.30000000000000019e-127 or 2.54999999999999992e-64 < F < 2.10000000000000006e-42

   1. Initial program 98.7%

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

      1. sqr-neg 98.7%

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

      2. associate-*l/ 99.0%

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

      3. *-lft-identity 99.0%

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

      4. sqr-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in l around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. unpow2 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.66 \cdot 10^{-139} \lor \neg \left(F \leq 2.3 \cdot 10^{-127} \lor \neg \left(F \leq 2.55 \cdot 10^{-64}\right) \land F \leq 2.1 \cdot 10^{-42}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 10: 92.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - \frac{1}{F \cdot F}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.8e+22) (not (<= l 1.25e-15)))
   (* PI l)
   (* l (* PI (- 1.0 (/ 1.0 (* F F)))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.8e+22) || !(l <= 1.25e-15)) {
		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 <= -1.8e+22) || !(l <= 1.25e-15)) {
		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 <= -1.8e+22) or not (l <= 1.25e-15):
		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 <= -1.8e+22) || !(l <= 1.25e-15))
		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 <= -1.8e+22) || ~((l <= 1.25e-15)))
		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, -1.8e+22], N[Not[LessEqual[l, 1.25e-15]], $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 < -1.8e22 or 1.25e-15 < l

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. associate-*l/ 58.5%

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

      3. *-lft-identity 58.5%

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

      4. sqr-neg 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in F around inf 97.4%

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

   if -1.8e22 < l < 1.25e-15

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. associate-*l/ 87.6%

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

      3. *-lft-identity 87.6%

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

      4. sqr-neg 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in l around 0 86.5%

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

      1. unpow2 86.5%

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

   6. Simplified 86.5%

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

      1. sub-neg 86.5%

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

      2. distribute-lft-in 86.5%

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

      3. *-commutative 86.5%

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

      4. div-inv 86.5%

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

      5. distribute-rgt-neg-in 86.5%

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

      6. pow2 86.5%

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

      7. pow-flip 86.5%

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

      8. metadata-eval 86.5%

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

   8. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. distribute-rgt-out 86.5%

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

      3. distribute-rgt-neg-out 86.5%

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

      4. sub-neg 86.5%

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

      5. *-rgt-identity 86.5%

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

      6. distribute-lft-out-- 86.5%

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

   10. Simplified 86.5%

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

   11. Taylor expanded in l around 0 86.5%

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

       1. unpow2 86.5%

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

   13. Simplified 86.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+22} \lor \neg \left(\ell \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - \frac{1}{F \cdot F}\right)\right)\\ \end{array} \]

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 71.9%

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

   1. sqr-neg 71.9%

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

   2. associate-*l/ 72.4%

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

   3. *-lft-identity 72.4%

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

   4. sqr-neg 72.4%

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

3. Simplified 72.4%

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

4. Taylor expanded in l around 0 64.6%

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

   1. unpow2 64.6%

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

6. Simplified 64.6%

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

7. Taylor expanded in F around inf 73.4%

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

8. Final simplification 73.4%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))