VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.8%

Time: 15.9s

Alternatives: 12

Speedup: 0.8×

• could not determine a ground truth

  1. F = -1.5768345251931338e+283
  2. l = -3351247144079930.5

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: 76.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		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+19) || !((Math.PI * l) <= 50000.0)) {
		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+19) or not ((math.pi * l) <= 50000.0):
		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+19) || !(Float64(pi * l) <= 50000.0))
		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+19) || ~(((pi * l) <= 50000.0)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $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) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

      1. associate-*l/ 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. associate-/r* 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F \cdot \frac{\frac{F}{\pi}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (+ (* PI l) (/ -1.0 (* F (/ (/ F PI) l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + (-1.0 / (F * ((F / ((double) M_PI)) / l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+19) || !((Math.PI * l) <= 50000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + (-1.0 / (F * ((F / Math.PI) / l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+19) or not ((math.pi * l) <= 50000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + (-1.0 / (F * ((F / math.pi) / l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+19) || !(Float64(pi * l) <= 50000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(F * Float64(Float64(F / pi) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+19) || ~(((pi * l) <= 50000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + (-1.0 / (F * ((F / pi) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F * N[(N[(F / Pi), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

      1. associate-/r* 99.0%

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

      2. div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in l around 0 97.0%

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

      1. clear-num 97.0%

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

      2. frac-times 97.0%

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

      3. metadata-eval 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-/r* 97.1%

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

   8. Applied egg-rr 97.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F \cdot \frac{\frac{F}{\pi}}{\ell}}\\ \end{array} \]

Alternative 3: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		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+19) || !((Math.PI * l) <= 50000.0)) {
		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+19) or not ((math.pi * l) <= 50000.0):
		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+19) || !(Float64(pi * l) <= 50000.0))
		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+19) || ~(((pi * l) <= 50000.0)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $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) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

   2. Taylor expanded in l around 0 87.3%

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

      1. associate-/l* 87.2%

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

      2. associate-/r/ 87.3%

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

      3. unpow2 87.3%

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

   4. Simplified 87.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		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) <= -5e+19) || !((Math.PI * l) <= 50000.0)) {
		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) <= -5e+19) or not ((math.pi * l) <= 50000.0):
		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) <= -5e+19) || !(Float64(pi * l) <= 50000.0))
		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) <= -5e+19) || ~(((pi * l) <= 50000.0)))
		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], -5e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $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) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

      1. associate-/r* 99.0%

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

      2. div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in l around 0 97.0%

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

   7. Taylor expanded in l around 0 87.3%

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

      1. *-commutative 87.3%

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

      2. unpow2 87.3%

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

      3. times-frac 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 5: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		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+19) || !((Math.PI * l) <= 50000.0)) {
		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+19) or not ((math.pi * l) <= 50000.0):
		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+19) || !(Float64(pi * l) <= 50000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+19) || ~(((pi * l) <= 50000.0)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $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) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

      1. associate-/r* 99.0%

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

      2. div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in l around 0 97.0%

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

      1. un-div-inv 97.0%

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

      2. associate-/l* 97.0%

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

      3. associate-/r/ 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. *-commutative 97.0%

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

   10. Applied egg-rr 97.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 6: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 50000.0)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 50000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+19) || !((Math.PI * l) <= 50000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+19) or not ((math.pi * l) <= 50000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+19) || !(Float64(pi * l) <= 50000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+19) || ~(((pi * l) <= 50000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e19 or 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-lft-identity 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in l around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e19 < (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.8%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in l around 0 86.2%

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

      1. unpow2 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 50000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 73.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(-{F}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-289)
   (* PI (* (/ l F) (/ -1.0 F)))
   (if (or (<= (* F F) 5e-90) (not (<= (* F F) 2e-41)))
     (* PI l)
     (* (* PI l) (- (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((double) M_PI) * ((l / F) * (-1.0 / F));
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) * -pow(F, -2.0);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = Math.PI * ((l / F) * (-1.0 / F));
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) * -Math.pow(F, -2.0);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-289:
		tmp = math.pi * ((l / F) * (-1.0 / F))
	elif ((F * F) <= 5e-90) or not ((F * F) <= 2e-41):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) * -math.pow(F, -2.0)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-289)
		tmp = Float64(pi * Float64(Float64(l / F) * Float64(-1.0 / F)));
	elseif ((Float64(F * F) <= 5e-90) || !(Float64(F * F) <= 2e-41))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) * Float64(-(F ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-289)
		tmp = pi * ((l / F) * (-1.0 / F));
	elseif (((F * F) <= 5e-90) || ~(((F * F) <= 2e-41)))
		tmp = pi * l;
	else
		tmp = (pi * l) * -(F ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-289], N[(Pi * N[(N[(l / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 5e-90], N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-41]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] * (-N[Power[F, -2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2e-289

   1. Initial program 39.2%

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

      1. associate-*l/ 40.0%

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

      2. *-lft-identity 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in l around 0 34.9%

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

      1. unpow2 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in F around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. associate-/r* 58.3%

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

       2. associate-*l/ 58.2%

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

       3. div-inv 58.3%

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

       4. *-commutative 58.3%

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

       5. associate-*l* 58.3%

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

   11. Applied egg-rr 58.3%

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

   if 2e-289 < (*.f64 F F) < 5.00000000000000019e-90 or 2.00000000000000001e-41 < (*.f64 F F)

   1. Initial program 89.5%

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

      1. associate-*l/ 89.5%

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

      2. *-lft-identity 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in l around 0 80.5%

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

      1. unpow2 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in F around inf 89.6%

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

   if 5.00000000000000019e-90 < (*.f64 F F) < 2.00000000000000001e-41

   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/ 99.5%

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

      2. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in F around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unpow2 72.3%

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

   9. Simplified 72.3%

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

       1. clear-num 72.3%

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

       2. associate-/r/ 72.6%

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

       3. pow2 72.6%

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

       4. pow-flip 72.7%

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

       5. metadata-eval 72.7%

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

       6. *-commutative 72.7%

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

   11. Applied egg-rr 72.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(-{F}^{-2}\right)\\ \end{array} \]

Alternative 8: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-289)
   (* (/ (/ l F) F) (- PI))
   (if (or (<= (* F F) 5e-90) (not (<= (* F F) 2e-41)))
     (* PI l)
     (* (/ l F) (/ (- PI) F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((l / F) / F) * -((double) M_PI);
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l / F) * (-((double) M_PI) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((l / F) / F) * -Math.PI;
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = Math.PI * l;
	} else {
		tmp = (l / F) * (-Math.PI / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-289:
		tmp = ((l / F) / F) * -math.pi
	elif ((F * F) <= 5e-90) or not ((F * F) <= 2e-41):
		tmp = math.pi * l
	else:
		tmp = (l / F) * (-math.pi / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-289)
		tmp = Float64(Float64(Float64(l / F) / F) * Float64(-pi));
	elseif ((Float64(F * F) <= 5e-90) || !(Float64(F * F) <= 2e-41))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l / F) * Float64(Float64(-pi) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-289)
		tmp = ((l / F) / F) * -pi;
	elseif (((F * F) <= 5e-90) || ~(((F * F) <= 2e-41)))
		tmp = pi * l;
	else
		tmp = (l / F) * (-pi / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-289], N[(N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision] * (-Pi)), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 5e-90], N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-41]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2e-289

   1. Initial program 39.2%

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

      1. associate-*l/ 40.0%

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

      2. *-lft-identity 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in l around 0 34.9%

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

      1. unpow2 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in F around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. associate-/r* 58.3%

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

       2. associate-*l/ 58.2%

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

       3. associate-/l* 58.3%

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

       4. associate-/r/ 58.3%

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

   11. Applied egg-rr 58.3%

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

   if 2e-289 < (*.f64 F F) < 5.00000000000000019e-90 or 2.00000000000000001e-41 < (*.f64 F F)

   1. Initial program 89.5%

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

      1. associate-*l/ 89.5%

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

      2. *-lft-identity 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in l around 0 80.5%

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

      1. unpow2 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in F around inf 89.6%

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

   if 5.00000000000000019e-90 < (*.f64 F F) < 2.00000000000000001e-41

   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/ 99.5%

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

      2. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in F around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unpow2 72.3%

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

   9. Simplified 72.3%

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

       1. *-commutative 72.3%

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

       2. times-frac 72.4%

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

   11. Applied egg-rr 72.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \]

Alternative 9: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-289)
   (* (/ (/ l F) F) (- PI))
   (if (or (<= (* F F) 5e-90) (not (<= (* F F) 2e-41)))
     (* PI l)
     (/ (- l) (/ (* F F) PI)))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((l / F) / F) * -((double) M_PI);
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = -l / ((F * F) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((l / F) / F) * -Math.PI;
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / ((F * F) / Math.PI);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-289:
		tmp = ((l / F) / F) * -math.pi
	elif ((F * F) <= 5e-90) or not ((F * F) <= 2e-41):
		tmp = math.pi * l
	else:
		tmp = -l / ((F * F) / math.pi)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-289)
		tmp = Float64(Float64(Float64(l / F) / F) * Float64(-pi));
	elseif ((Float64(F * F) <= 5e-90) || !(Float64(F * F) <= 2e-41))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-289)
		tmp = ((l / F) / F) * -pi;
	elseif (((F * F) <= 5e-90) || ~(((F * F) <= 2e-41)))
		tmp = pi * l;
	else
		tmp = -l / ((F * F) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-289], N[(N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision] * (-Pi)), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 5e-90], N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-41]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2e-289

   1. Initial program 39.2%

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

      1. associate-*l/ 40.0%

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

      2. *-lft-identity 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in l around 0 34.9%

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

      1. unpow2 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in F around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. associate-/r* 58.3%

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

       2. associate-*l/ 58.2%

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

       3. associate-/l* 58.3%

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

       4. associate-/r/ 58.3%

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

   11. Applied egg-rr 58.3%

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

   if 2e-289 < (*.f64 F F) < 5.00000000000000019e-90 or 2.00000000000000001e-41 < (*.f64 F F)

   1. Initial program 89.5%

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

      1. associate-*l/ 89.5%

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

      2. *-lft-identity 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in l around 0 80.5%

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

      1. unpow2 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in F around inf 89.6%

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

   if 5.00000000000000019e-90 < (*.f64 F F) < 2.00000000000000001e-41

   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/ 99.5%

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

      2. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in F around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unpow2 72.3%

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

      3. associate-/l* 72.6%

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

      4. distribute-neg-frac 72.6%

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

   9. Simplified 72.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 10: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-289)
   (* PI (* (/ l F) (/ -1.0 F)))
   (if (or (<= (* F F) 5e-90) (not (<= (* F F) 2e-41)))
     (* PI l)
     (/ (- l) (/ (* F F) PI)))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = ((double) M_PI) * ((l / F) * (-1.0 / F));
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = -l / ((F * F) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-289) {
		tmp = Math.PI * ((l / F) * (-1.0 / F));
	} else if (((F * F) <= 5e-90) || !((F * F) <= 2e-41)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / ((F * F) / Math.PI);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-289:
		tmp = math.pi * ((l / F) * (-1.0 / F))
	elif ((F * F) <= 5e-90) or not ((F * F) <= 2e-41):
		tmp = math.pi * l
	else:
		tmp = -l / ((F * F) / math.pi)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-289)
		tmp = Float64(pi * Float64(Float64(l / F) * Float64(-1.0 / F)));
	elseif ((Float64(F * F) <= 5e-90) || !(Float64(F * F) <= 2e-41))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-289)
		tmp = pi * ((l / F) * (-1.0 / F));
	elseif (((F * F) <= 5e-90) || ~(((F * F) <= 2e-41)))
		tmp = pi * l;
	else
		tmp = -l / ((F * F) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-289], N[(Pi * N[(N[(l / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 5e-90], N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-41]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2e-289

   1. Initial program 39.2%

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

      1. associate-*l/ 40.0%

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

      2. *-lft-identity 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in l around 0 34.9%

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

      1. unpow2 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in F around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. associate-/r* 58.3%

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

       2. associate-*l/ 58.2%

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

       3. div-inv 58.3%

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

       4. *-commutative 58.3%

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

       5. associate-*l* 58.3%

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

   11. Applied egg-rr 58.3%

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

   if 2e-289 < (*.f64 F F) < 5.00000000000000019e-90 or 2.00000000000000001e-41 < (*.f64 F F)

   1. Initial program 89.5%

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

      1. associate-*l/ 89.5%

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

      2. *-lft-identity 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in l around 0 80.5%

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

      1. unpow2 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in F around inf 89.6%

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

   if 5.00000000000000019e-90 < (*.f64 F F) < 2.00000000000000001e-41

   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/ 99.5%

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

      2. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in F around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unpow2 72.3%

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

      3. associate-/l* 72.6%

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

      4. distribute-neg-frac 72.6%

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

   9. Simplified 72.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-90} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 11: 50.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 4.2 \cdot 10^{-145} \lor \neg \left(F \leq 4.4 \cdot 10^{-45}\right) \land F \leq 5.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 4.2e-145) (and (not (<= F 4.4e-45)) (<= F 5.8e-20)))
   (* (/ l F) (/ (- PI) F))
   (* PI l)))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 4.2e-145) || (!(F <= 4.4e-45) && (F <= 5.8e-20))) {
		tmp = (l / F) * (-((double) M_PI) / F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 4.2e-145) || (!(F <= 4.4e-45) && (F <= 5.8e-20))) {
		tmp = (l / F) * (-Math.PI / F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 4.2e-145) or (not (F <= 4.4e-45) and (F <= 5.8e-20)):
		tmp = (l / F) * (-math.pi / F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 4.2e-145) || (!(F <= 4.4e-45) && (F <= 5.8e-20)))
		tmp = Float64(Float64(l / F) * Float64(Float64(-pi) / F));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 4.2e-145) || (~((F <= 4.4e-45)) && (F <= 5.8e-20)))
		tmp = (l / F) * (-pi / F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 4.2e-145], And[N[Not[LessEqual[F, 4.4e-45]], $MachinePrecision], LessEqual[F, 5.8e-20]]], N[(N[(l / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.19999999999999982e-145 or 4.39999999999999987e-45 < F < 5.8e-20

   1. Initial program 72.5%

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

      1. associate-*l/ 72.8%

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

      2. *-lft-identity 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in l around 0 64.0%

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

      1. unpow2 64.0%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in F around 0 22.2%

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

      1. mul-1-neg 22.2%

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

      2. unpow2 22.2%

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

   9. Simplified 22.2%

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

       1. *-commutative 22.2%

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

       2. times-frac 29.3%

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

   11. Applied egg-rr 29.3%

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

   if 4.19999999999999982e-145 < F < 4.39999999999999987e-45 or 5.8e-20 < F

   1. Initial program 90.6%

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

      1. associate-*l/ 90.6%

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

      2. *-lft-identity 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in l around 0 81.5%

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

      1. unpow2 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in F around inf 87.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.2 \cdot 10^{-145} \lor \neg \left(F \leq 4.4 \cdot 10^{-45}\right) \land F \leq 5.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 12: 73.2% 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 78.1%

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

   1. associate-*l/ 78.3%

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

   2. *-lft-identity 78.3%

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

3. Simplified 78.3%

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

4. Taylor expanded in l around 0 69.5%

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

   1. unpow2 69.5%

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

6. Simplified 69.5%

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

7. Taylor expanded in F around inf 75.5%

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

8. Final simplification 75.5%

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

Reproduce

herbie shell --seed 2023249 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))