VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.0% → 98.4%

Time: 17.2s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = 3.321124107723376e+126
  2. l = -1868562082141109.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% 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: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		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) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		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) <= -2e+24) or not ((math.pi * l) <= 0.2):
		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) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		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) <= -2e+24) || ~(((pi * l) <= 0.2)))
		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], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $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) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		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) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		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) <= -2e+24) or not ((math.pi * l) <= 0.2):
		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) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		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) <= -2e+24) || ~(((pi * l) <= 0.2)))
		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], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $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) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in l around 0 92.0%

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

      1. associate-/l* 92.0%

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

      2. associate-/r/ 92.0%

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

      3. unpow2 92.0%

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

   4. Simplified 92.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (/ l (/ (* F F) PI)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (l / ((F * F) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (l / ((F * F) / Math.PI));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+24) or not ((math.pi * l) <= 0.2):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (l / ((F * F) / math.pi))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(l / Float64(Float64(F * F) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+24) || ~(((pi * l) <= 0.2)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (l / ((F * F) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(l / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in l around 0 92.0%

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

      1. associate-/l* 92.0%

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

      2. unpow2 92.0%

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

   4. Simplified 92.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 4: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (/ PI (* F (/ F l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F * (F / l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI / (F * (F / l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+24) or not ((math.pi * l) <= 0.2):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi / (F * (F / l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F * Float64(F / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+24) || ~(((pi * l) <= 0.2)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi / (F * (F / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-/l* 92.0%

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

      3. unpow2 92.0%

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

      4. associate-*r/ 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		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) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		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) <= -2e+24) or not ((math.pi * l) <= 0.2):
		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) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * Float64(l / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+24) || ~(((pi * l) <= 0.2)))
		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], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in l around 0 92.0%

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

      1. associate-/l* 92.0%

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

      2. associate-/r/ 92.0%

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

      3. unpow2 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 98.7%

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

      2. associate-*l/ 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 6: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		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) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		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) <= -2e+24) or not ((math.pi * l) <= 0.2):
		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) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		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) <= -2e+24) || ~(((pi * l) <= 0.2)))
		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], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $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) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in l around 0 92.0%

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

      1. associate-/l* 92.0%

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

      2. associate-/r/ 92.0%

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

      3. unpow2 92.0%

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

   4. Simplified 92.0%

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

      1. associate-*l/ 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-/r* 98.8%

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

   6. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 7: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+24) (not (<= (* PI l) 0.2)))
   (* PI l)
   (* PI (- l (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+24) || !((((double) M_PI) * l) <= 0.2)) {
		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) <= -2e+24) || !((Math.PI * l) <= 0.2)) {
		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) <= -2e+24) or not ((math.pi * l) <= 0.2):
		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) <= -2e+24) || !(Float64(pi * l) <= 0.2))
		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) <= -2e+24) || ~(((pi * l) <= 0.2)))
		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], -2e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.2]], $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) < -2e24 or 0.20000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 51.3%

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

      1. associate-/l* 51.3%

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

      2. associate-/r/ 51.3%

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

      3. unpow2 51.3%

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

   4. Simplified 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l/ 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -2e24 < (*.f64 (PI.f64) l) < 0.20000000000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in l around 0 92.0%

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

      1. associate-/l* 92.0%

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

      2. associate-/r/ 92.0%

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

      3. unpow2 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 98.7%

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

      2. associate-*l/ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in l around 0 91.2%

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

      1. sub-neg 91.2%

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

      2. mul-1-neg 91.2%

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

      3. distribute-rgt-in 91.2%

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

      4. associate-*r/ 91.2%

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

      5. unpow2 91.2%

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

      6. times-frac 91.2%

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

      7. metadata-eval 91.2%

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

      8. distribute-neg-frac 91.2%

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

      9. associate-*r* 98.7%

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

      10. associate-*l/ 98.7%

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

      11. associate-*r/ 98.7%

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

      12. cancel-sign-sub-inv 98.7%

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

      13. *-commutative 98.7%

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

      14. associate-*r* 98.7%

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

      15. distribute-lft-out-- 98.7%

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

      16. associate-*r/ 98.7%

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

   9. Simplified 92.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 0.2\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \]

Alternative 8: 73.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in l around 0 70.7%

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

   1. associate-/l* 70.7%

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

   2. associate-/r/ 70.7%

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

   3. unpow2 70.7%

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

4. Simplified 70.7%

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

   1. associate-/r* 73.9%

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

   2. associate-*l/ 73.9%

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

6. Applied egg-rr 73.9%

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

7. Taylor expanded in F around inf 80.2%

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

8. Final simplification 80.2%

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

Reproduce

herbie shell --seed 2023287 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))