VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 98.5%

Time: 23.7s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+34) || !((((double) M_PI) * l) <= 2e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -1e+34) || !((Math.PI * l) <= 2e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e+34) or not ((math.pi * l) <= 2e+15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+34) || !(Float64(pi * l) <= 2e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e+34) || ~(((pi * l) <= 2e+15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+34], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 62.6%

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

      1. sqr-neg 62.6%

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

      2. associate-*l/ 62.6%

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

      3. *-lft-identity 62.6%

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

      4. sqr-neg 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in l around 0 47.1%

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

      1. unpow2 47.1%

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

   6. Simplified 47.1%

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

   7. Taylor expanded in F around inf 99.5%

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

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

   1. Initial program 90.0%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

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

   1. Initial program 90.0%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in l around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. unpow2 90.6%

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

      3. times-frac 98.8%

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

   6. 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%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

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

   1. Initial program 90.0%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in l around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. associate-*l/ 63.2%

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

      3. *-lft-identity 63.2%

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

      4. sqr-neg 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in F around inf 98.8%

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

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

   1. Initial program 90.0%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in l around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. unpow2 90.6%

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

      3. times-frac 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in l around 0 89.8%

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

      1. distribute-rgt-out-- 89.8%

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

      2. *-commutative 89.8%

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

      3. unpow2 89.8%

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

      4. associate-/r/ 90.5%

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

      5. associate-/l* 98.8%

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

      6. associate-/l* 98.8%

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

      7. *-commutative 98.8%

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

      8. associate-*l/ 98.8%

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

      9. associate-/r* 90.6%

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

      10. distribute-rgt-out-- 90.6%

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

   9. Simplified 90.6%

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

4. Final simplification 95.1%

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

Alternative 5: 62.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \ell + -1\\ t_1 := \frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{if}\;F \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (+ (* PI l) -1.0)) (t_1 (* (/ (/ l F) F) (- PI))))
   (if (<= F 2.1e-107)
     t_0
     (if (<= F 2e-83)
       t_1
       (if (<= F 8.5e-38) t_0 (if (<= F 2e-17) t_1 (* PI l)))))))

C

double code(double F, double l) {
	double t_0 = (((double) M_PI) * l) + -1.0;
	double t_1 = ((l / F) / F) * -((double) M_PI);
	double tmp;
	if (F <= 2.1e-107) {
		tmp = t_0;
	} else if (F <= 2e-83) {
		tmp = t_1;
	} else if (F <= 8.5e-38) {
		tmp = t_0;
	} else if (F <= 2e-17) {
		tmp = t_1;
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = (Math.PI * l) + -1.0;
	double t_1 = ((l / F) / F) * -Math.PI;
	double tmp;
	if (F <= 2.1e-107) {
		tmp = t_0;
	} else if (F <= 2e-83) {
		tmp = t_1;
	} else if (F <= 8.5e-38) {
		tmp = t_0;
	} else if (F <= 2e-17) {
		tmp = t_1;
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = (math.pi * l) + -1.0
	t_1 = ((l / F) / F) * -math.pi
	tmp = 0
	if F <= 2.1e-107:
		tmp = t_0
	elif F <= 2e-83:
		tmp = t_1
	elif F <= 8.5e-38:
		tmp = t_0
	elif F <= 2e-17:
		tmp = t_1
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(pi * l) + -1.0)
	t_1 = Float64(Float64(Float64(l / F) / F) * Float64(-pi))
	tmp = 0.0
	if (F <= 2.1e-107)
		tmp = t_0;
	elseif (F <= 2e-83)
		tmp = t_1;
	elseif (F <= 8.5e-38)
		tmp = t_0;
	elseif (F <= 2e-17)
		tmp = t_1;
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = (pi * l) + -1.0;
	t_1 = ((l / F) / F) * -pi;
	tmp = 0.0;
	if (F <= 2.1e-107)
		tmp = t_0;
	elseif (F <= 2e-83)
		tmp = t_1;
	elseif (F <= 8.5e-38)
		tmp = t_0;
	elseif (F <= 2e-17)
		tmp = t_1;
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision] * (-Pi)), $MachinePrecision]}, If[LessEqual[F, 2.1e-107], t$95$0, If[LessEqual[F, 2e-83], t$95$1, If[LessEqual[F, 8.5e-38], t$95$0, If[LessEqual[F, 2e-17], t$95$1, N[(Pi * l), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.0999999999999999e-107 or 2.0000000000000001e-83 < F < 8.50000000000000046e-38

   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-/r/ 64.6%

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

      2. inv-pow 64.6%

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

      3. pow-to-exp 30.5%

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

      4. associate-/l* 33.5%

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

   3. Applied egg-rr 33.5%

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

      1. exp-to-pow 70.1%

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

      2. inv-pow 70.1%

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

      3. add-sqr-sqrt 34.0%

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

      4. associate-/r* 34.0%

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

      5. metadata-eval 34.0%

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

      6. sqrt-div 33.9%

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

      7. clear-num 33.9%

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

      8. associate-/r* 30.6%

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

      9. sqrt-div 30.6%

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

      10. sqrt-prod 10.2%

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

      11. add-sqr-sqrt 27.1%

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

      12. add-exp-log 26.9%

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

      13. add-sqr-sqrt 15.5%

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

   5. Applied egg-rr 31.5%

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

      1. *-inverses 57.7%

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

   7. Simplified 57.7%

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

   if 2.0999999999999999e-107 < F < 2.0000000000000001e-83 or 8.50000000000000046e-38 < F < 2.00000000000000014e-17

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in l around 0 79.6%

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

      1. *-commutative 79.6%

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

      2. unpow2 79.6%

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

      3. times-frac 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in F around 0 79.6%

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

      1. mul-1-neg 79.6%

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

      2. unpow2 79.6%

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

      3. associate-*l/ 80.0%

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

      4. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

   if 2.00000000000000014e-17 < F

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

      4. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 97.9%

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

      1. unpow2 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in F around inf 99.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6: 62.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \ell + -1\\ \mathbf{if}\;F \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (+ (* PI l) -1.0)))
   (if (<= F 6.2e-109)
     t_0
     (if (<= F 2.05e-82)
       (/ (- l) (/ (* F F) PI))
       (if (<= F 9.8e-38)
         t_0
         (if (<= F 2.2e-17) (* (/ (/ l F) F) (- PI)) (* PI l)))))))

C

double code(double F, double l) {
	double t_0 = (((double) M_PI) * l) + -1.0;
	double tmp;
	if (F <= 6.2e-109) {
		tmp = t_0;
	} else if (F <= 2.05e-82) {
		tmp = -l / ((F * F) / ((double) M_PI));
	} else if (F <= 9.8e-38) {
		tmp = t_0;
	} else if (F <= 2.2e-17) {
		tmp = ((l / F) / F) * -((double) M_PI);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = (Math.PI * l) + -1.0;
	double tmp;
	if (F <= 6.2e-109) {
		tmp = t_0;
	} else if (F <= 2.05e-82) {
		tmp = -l / ((F * F) / Math.PI);
	} else if (F <= 9.8e-38) {
		tmp = t_0;
	} else if (F <= 2.2e-17) {
		tmp = ((l / F) / F) * -Math.PI;
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = (math.pi * l) + -1.0
	tmp = 0
	if F <= 6.2e-109:
		tmp = t_0
	elif F <= 2.05e-82:
		tmp = -l / ((F * F) / math.pi)
	elif F <= 9.8e-38:
		tmp = t_0
	elif F <= 2.2e-17:
		tmp = ((l / F) / F) * -math.pi
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(pi * l) + -1.0)
	tmp = 0.0
	if (F <= 6.2e-109)
		tmp = t_0;
	elseif (F <= 2.05e-82)
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	elseif (F <= 9.8e-38)
		tmp = t_0;
	elseif (F <= 2.2e-17)
		tmp = Float64(Float64(Float64(l / F) / F) * Float64(-pi));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = (pi * l) + -1.0;
	tmp = 0.0;
	if (F <= 6.2e-109)
		tmp = t_0;
	elseif (F <= 2.05e-82)
		tmp = -l / ((F * F) / pi);
	elseif (F <= 9.8e-38)
		tmp = t_0;
	elseif (F <= 2.2e-17)
		tmp = ((l / F) / F) * -pi;
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[F, 6.2e-109], t$95$0, If[LessEqual[F, 2.05e-82], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.8e-38], t$95$0, If[LessEqual[F, 2.2e-17], N[(N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision] * (-Pi)), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 6.1999999999999999e-109 or 2.04999999999999998e-82 < F < 9.80000000000000078e-38

   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-/r/ 64.6%

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

      2. inv-pow 64.6%

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

      3. pow-to-exp 30.5%

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

      4. associate-/l* 33.5%

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

   3. Applied egg-rr 33.5%

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

      1. exp-to-pow 70.1%

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

      2. inv-pow 70.1%

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

      3. add-sqr-sqrt 34.0%

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

      4. associate-/r* 34.0%

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

      5. metadata-eval 34.0%

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

      6. sqrt-div 33.9%

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

      7. clear-num 33.9%

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

      8. associate-/r* 30.6%

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

      9. sqrt-div 30.6%

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

      10. sqrt-prod 10.2%

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

      11. add-sqr-sqrt 27.1%

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

      12. add-exp-log 26.9%

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

      13. add-sqr-sqrt 15.5%

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

   5. Applied egg-rr 31.5%

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

      1. *-inverses 57.7%

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

   7. Simplified 57.7%

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

   if 6.1999999999999999e-109 < F < 2.04999999999999998e-82

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. add-cbrt-cube 79.1%

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

      2. pow3 79.1%

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

   8. Applied egg-rr 79.1%

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

   9. Taylor expanded in F around 0 79.4%

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

       1. mul-1-neg 79.4%

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

       2. associate-/l* 80.0%

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

       3. unpow2 80.0%

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

   11. Simplified 80.0%

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

   if 9.80000000000000078e-38 < F < 2.2e-17

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. times-frac 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in F around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. unpow2 79.7%

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

      3. associate-*l/ 80.0%

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

      4. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

   if 2.2e-17 < F

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

      4. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 97.9%

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

      1. unpow2 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in F around inf 99.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-38}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7: 74.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sqr-neg 75.3%

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

   2. associate-*l/ 75.7%

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

   3. *-lft-identity 75.7%

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

   4. sqr-neg 75.7%

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

3. Simplified 75.7%

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

4. Taylor expanded in l around 0 66.7%

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

   1. unpow2 66.7%

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

6. Simplified 66.7%

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

7. Taylor expanded in F around inf 76.5%

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

8. Final simplification 76.5%

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

Reproduce

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