VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 98.8%

Time: 13.9s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = -1.143681103955322e-21
  2. l = -3560348628620921.5

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: 76.6% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 1000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1000.0)) {
		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+17) || !((Math.PI * l) <= 1000.0)) {
		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+17) or not ((math.pi * l) <= 1000.0):
		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+17) || !(Float64(pi * l) <= 1000.0))
		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+17) || ~(((pi * l) <= 1000.0)))
		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+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000.0]], $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) < -5e17 or 1e3 < (*.f64 (PI.f64) l)

   1. Initial program 55.8%

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

      1. associate-*l/ 55.8%

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

      2. *-lft-identity 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in l around 0 44.1%

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

      1. unpow2 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1e3

   1. Initial program 94.9%

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

      1. associate-*l/ 95.1%

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

      2. *-un-lft-identity 95.1%

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

      3. associate-/r* 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 1000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1e-5)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1e-5)) {
		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+17) || !((Math.PI * l) <= 1e-5)) {
		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+17) or not ((math.pi * l) <= 1e-5):
		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+17) || !(Float64(pi * l) <= 1e-5))
		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 (((pi * l) <= -5e+17) || ~(((pi * l) <= 1e-5)))
		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+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-5]], $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 (*.f64 (PI.f64) l) < -5e17 or 1.00000000000000008e-5 < (*.f64 (PI.f64) l)

   1. Initial program 56.5%

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

      1. associate-*l/ 56.5%

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

      2. *-lft-identity 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 44.7%

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

      1. unpow2 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1.00000000000000008e-5

   1. Initial program 94.8%

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

   2. Taylor expanded in l around 0 94.8%

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

      1. associate-/l* 94.8%

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

      2. associate-/r/ 94.9%

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

      3. unpow2 94.9%

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

   4. Simplified 94.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1e-5)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / (F / ((double) M_PI))) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+17) || !((Math.PI * l) <= 1e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / (F / Math.PI)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+17) or not ((math.pi * l) <= 1e-5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / (F / math.pi)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+17) || !(Float64(pi * l) <= 1e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / Float64(F / pi)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+17) || ~(((pi * l) <= 1e-5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / (F / pi)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e17 or 1.00000000000000008e-5 < (*.f64 (PI.f64) l)

   1. Initial program 56.5%

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

      1. associate-*l/ 56.5%

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

      2. *-lft-identity 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 44.7%

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

      1. unpow2 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1.00000000000000008e-5

   1. Initial program 94.8%

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

      1. associate-*l/ 95.0%

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

      2. *-un-lft-identity 95.0%

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

      3. associate-/r* 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in l around 0 99.5%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 4: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1e-5)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1e-5)) {
		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 (((Math.PI * l) <= -5e+17) || !((Math.PI * l) <= 1e-5)) {
		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 ((math.pi * l) <= -5e+17) or not ((math.pi * l) <= 1e-5):
		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 ((Float64(pi * l) <= -5e+17) || !(Float64(pi * l) <= 1e-5))
		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 (((pi * l) <= -5e+17) || ~(((pi * l) <= 1e-5)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-5]], $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 (*.f64 (PI.f64) l) < -5e17 or 1.00000000000000008e-5 < (*.f64 (PI.f64) l)

   1. Initial program 56.5%

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

      1. associate-*l/ 56.5%

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

      2. *-lft-identity 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 44.7%

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

      1. unpow2 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1.00000000000000008e-5

   1. Initial program 94.8%

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

      1. associate-*l/ 95.0%

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

      2. *-lft-identity 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in l around 0 94.6%

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

      1. unpow2 94.6%

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

   6. Simplified 94.6%

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

      1. sub-neg 94.6%

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

      2. distribute-lft-in 94.6%

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

      3. *-commutative 94.6%

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

      4. div-inv 94.6%

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

      5. distribute-lft-neg-in 94.6%

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

      6. pow2 94.6%

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

      7. pow-flip 94.6%

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

      8. metadata-eval 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. *-commutative 94.6%

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

      2. distribute-lft-in 94.6%

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

      3. cancel-sign-sub-inv 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 94.6%

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

   10. Simplified 94.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + \frac{-1}{F \cdot F}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1e-5)))
   (* PI l)
   (* l (* PI (+ 1.0 (/ -1.0 (* F F)))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1e-5)) {
		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 (((Math.PI * l) <= -5e+17) || !((Math.PI * l) <= 1e-5)) {
		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 ((math.pi * l) <= -5e+17) or not ((math.pi * l) <= 1e-5):
		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 ((Float64(pi * l) <= -5e+17) || !(Float64(pi * l) <= 1e-5))
		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 (((pi * l) <= -5e+17) || ~(((pi * l) <= 1e-5)))
		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[N[(Pi * l), $MachinePrecision], -5e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-5]], $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 (*.f64 (PI.f64) l) < -5e17 or 1.00000000000000008e-5 < (*.f64 (PI.f64) l)

   1. Initial program 56.5%

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

      1. associate-*l/ 56.5%

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

      2. *-lft-identity 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 44.7%

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

      1. unpow2 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1.00000000000000008e-5

   1. Initial program 94.8%

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

      1. associate-*l/ 95.0%

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

      2. *-lft-identity 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in l around 0 94.6%

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

      1. unpow2 94.6%

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

   6. Simplified 94.6%

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

      1. sub-neg 94.6%

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

      2. distribute-lft-in 94.6%

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

      3. *-commutative 94.6%

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

      4. div-inv 94.6%

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

      5. distribute-lft-neg-in 94.6%

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

      6. pow2 94.6%

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

      7. pow-flip 94.6%

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

      8. metadata-eval 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. *-commutative 94.6%

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

      2. distribute-lft-in 94.6%

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

      3. cancel-sign-sub-inv 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in l around 0 94.6%

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

       1. *-commutative 94.6%

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

       2. unpow2 94.6%

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

   13. Simplified 94.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + \frac{-1}{F \cdot F}\right)\right)\\ \end{array} \]

Alternative 6: 50.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (/ PI F) (/ (- l) F))))
   (if (<= F 1.25e-271)
     t_0
     (if (<= F 1.45e-108)
       (+ (* PI l) -1.0)
       (if (<= F 4.2e-15) t_0 (* PI l))))))

C

double code(double F, double l) {
	double t_0 = (((double) M_PI) / F) * (-l / F);
	double tmp;
	if (F <= 1.25e-271) {
		tmp = t_0;
	} else if (F <= 1.45e-108) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if (F <= 4.2e-15) {
		tmp = t_0;
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = (Math.PI / F) * (-l / F);
	double tmp;
	if (F <= 1.25e-271) {
		tmp = t_0;
	} else if (F <= 1.45e-108) {
		tmp = (Math.PI * l) + -1.0;
	} else if (F <= 4.2e-15) {
		tmp = t_0;
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = (math.pi / F) * (-l / F)
	tmp = 0
	if F <= 1.25e-271:
		tmp = t_0
	elif F <= 1.45e-108:
		tmp = (math.pi * l) + -1.0
	elif F <= 4.2e-15:
		tmp = t_0
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(pi / F) * Float64(Float64(-l) / F))
	tmp = 0.0
	if (F <= 1.25e-271)
		tmp = t_0;
	elseif (F <= 1.45e-108)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (F <= 4.2e-15)
		tmp = t_0;
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = (pi / F) * (-l / F);
	tmp = 0.0;
	if (F <= 1.25e-271)
		tmp = t_0;
	elseif (F <= 1.45e-108)
		tmp = (pi * l) + -1.0;
	elseif (F <= 4.2e-15)
		tmp = t_0;
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 1.25e-271], t$95$0, If[LessEqual[F, 1.45e-108], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[F, 4.2e-15], t$95$0, N[(Pi * l), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.2500000000000001e-271 or 1.45e-108 < F < 4.19999999999999962e-15

   1. Initial program 74.4%

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

      1. associate-*l/ 74.4%

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

      2. *-lft-identity 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in l around 0 66.3%

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

      1. unpow2 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in F around 0 30.2%

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

      1. associate-*r/ 30.2%

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

      2. neg-mul-1 30.2%

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

      3. unpow2 30.2%

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

   9. Simplified 30.2%

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

   10. Taylor expanded in l around 0 30.2%

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

       1. mul-1-neg 30.2%

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

       2. *-commutative 30.2%

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

       3. unpow2 30.2%

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

       4. times-frac 32.6%

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

   12. Simplified 32.6%

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

   if 1.2500000000000001e-271 < F < 1.45e-108

   1. Initial program 36.7%

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

      1. associate-/r/ 37.6%

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

      2. inv-pow 37.6%

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

      3. pow-to-exp 9.7%

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

      4. associate-/l* 14.6%

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

   3. Applied egg-rr 14.6%

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

   5. Applied egg-rr 31.4%

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

      1. *-inverses 59.4%

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

   7. Simplified 59.4%

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

   if 4.19999999999999962e-15 < F

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   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 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 inf 93.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7: 50.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\ell}{F}\\ \mathbf{if}\;F \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{\pi}{F} \cdot t_0\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\pi \cdot \frac{t_0}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (/ (- l) F)))
   (if (<= F 9.5e-272)
     (* (/ PI F) t_0)
     (if (<= F 1.35e-108)
       (+ (* PI l) -1.0)
       (if (<= F 1.7e-14) (* PI (/ t_0 F)) (* PI l))))))

C

double code(double F, double l) {
	double t_0 = -l / F;
	double tmp;
	if (F <= 9.5e-272) {
		tmp = (((double) M_PI) / F) * t_0;
	} else if (F <= 1.35e-108) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if (F <= 1.7e-14) {
		tmp = ((double) M_PI) * (t_0 / F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = -l / F;
	double tmp;
	if (F <= 9.5e-272) {
		tmp = (Math.PI / F) * t_0;
	} else if (F <= 1.35e-108) {
		tmp = (Math.PI * l) + -1.0;
	} else if (F <= 1.7e-14) {
		tmp = Math.PI * (t_0 / F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = -l / F
	tmp = 0
	if F <= 9.5e-272:
		tmp = (math.pi / F) * t_0
	elif F <= 1.35e-108:
		tmp = (math.pi * l) + -1.0
	elif F <= 1.7e-14:
		tmp = math.pi * (t_0 / F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(-l) / F)
	tmp = 0.0
	if (F <= 9.5e-272)
		tmp = Float64(Float64(pi / F) * t_0);
	elseif (F <= 1.35e-108)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (F <= 1.7e-14)
		tmp = Float64(pi * Float64(t_0 / F));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = -l / F;
	tmp = 0.0;
	if (F <= 9.5e-272)
		tmp = (pi / F) * t_0;
	elseif (F <= 1.35e-108)
		tmp = (pi * l) + -1.0;
	elseif (F <= 1.7e-14)
		tmp = pi * (t_0 / F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[((-l) / F), $MachinePrecision]}, If[LessEqual[F, 9.5e-272], N[(N[(Pi / F), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[F, 1.35e-108], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[F, 1.7e-14], N[(Pi * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 9.50000000000000024e-272

   1. Initial program 75.2%

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

      1. associate-*l/ 75.2%

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

      2. *-lft-identity 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in l around 0 67.0%

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

      1. unpow2 67.0%

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

   6. Simplified 67.0%

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

   7. Taylor expanded in F around 0 27.2%

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

      1. associate-*r/ 27.2%

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

      2. neg-mul-1 27.2%

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

      3. unpow2 27.2%

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

   9. Simplified 27.2%

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

   10. Taylor expanded in l around 0 27.1%

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

       1. mul-1-neg 27.1%

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

       2. *-commutative 27.1%

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

       3. unpow2 27.1%

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

       4. times-frac 29.7%

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

   12. Simplified 29.7%

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

   if 9.50000000000000024e-272 < F < 1.35000000000000002e-108

   1. Initial program 36.7%

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

      1. associate-/r/ 37.6%

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

      2. inv-pow 37.6%

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

      3. pow-to-exp 9.7%

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

      4. associate-/l* 14.6%

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

   3. Applied egg-rr 14.6%

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

   5. Applied egg-rr 31.4%

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

      1. *-inverses 59.4%

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

   7. Simplified 59.4%

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

   if 1.35000000000000002e-108 < F < 1.70000000000000001e-14

   1. Initial program 66.7%

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

      1. associate-*l/ 66.9%

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

      2. *-lft-identity 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in l around 0 59.8%

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

      1. unpow2 59.8%

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

   6. Simplified 59.8%

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

   7. Taylor expanded in F around 0 60.0%

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

      1. mul-1-neg 60.0%

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

      2. associate-/l* 59.9%

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

      3. unpow2 59.9%

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

      4. associate-*r/ 59.8%

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

   9. Simplified 59.8%

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

       1. associate-/r* 60.0%

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

       2. associate-/r/ 60.0%

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

   11. Applied egg-rr 60.0%

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

   if 1.70000000000000001e-14 < F

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   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 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 inf 93.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\pi \cdot \frac{\frac{-\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8: 74.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. associate-*l/ 77.0%

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

   2. *-lft-identity 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in l around 0 71.2%

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

   1. unpow2 71.2%

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

6. Simplified 71.2%

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

7. Taylor expanded in F around inf 74.3%

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

8. Final simplification 74.3%

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

Reproduce

herbie shell --seed 2023227 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))