VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 98.5%

Time: 14.3s

Alternatives: 11

Speedup: 1.5×

• could not determine a ground truth

  1. F = 1.309265746103383e-25
  2. l = 1461691357502982.5

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: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+31} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{F}}}{\sqrt{F}}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+31) (not (<= (* PI l) 1000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (/ (tan (* PI l)) F) (sqrt F)) (sqrt F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+31) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((tan((((double) M_PI) * l)) / F) / sqrt(F)) / sqrt(F));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+31) || !((Math.PI * l) <= 1000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (((Math.tan((Math.PI * l)) / F) / Math.sqrt(F)) / Math.sqrt(F));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+31) or not ((math.pi * l) <= 1000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (((math.tan((math.pi * l)) / F) / math.sqrt(F)) / math.sqrt(F))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+31) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(tan(Float64(pi * l)) / F) / sqrt(F)) / sqrt(F)));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+31) || ~(((pi * l) <= 1000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (((tan((pi * l)) / F) / sqrt(F)) / sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+31], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1.9999999999999999e31 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 61.7%

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

      1. associate-*l/ 61.7%

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

      2. *-lft-identity 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in l around 0 45.7%

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

      1. unpow2 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -1.9999999999999999e31 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 87.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-un-lft-identity 89.5%

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

      3. associate-/r* 98.7%

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

      4. add-sqr-sqrt 52.7%

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

      5. associate-/r* 52.7%

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

   3. Applied egg-rr 52.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+31} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{F}}}{\sqrt{F}}\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+31} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+31) (not (<= (* PI l) 1000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+31) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+31) || !((Math.PI * l) <= 1000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+31) or not ((math.pi * l) <= 1000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+31) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+31) || ~(((pi * l) <= 1000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+31], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $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) < -1.9999999999999999e31 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 61.7%

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

      1. associate-*l/ 61.7%

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

      2. *-lft-identity 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in l around 0 45.7%

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

      1. unpow2 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -1.9999999999999999e31 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 87.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-un-lft-identity 89.5%

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

      3. associate-/r* 98.7%

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

   3. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+31} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000000 \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -100000000.0) (not (<= (* PI l) 1000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (* l (/ PI F)) F))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -100000000.0) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l * (((double) M_PI) / F)) / F);
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -100000000.0) || !((Math.PI * l) <= 1000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l * (Math.PI / F)) / F);
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -100000000.0) or not ((math.pi * l) <= 1000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l * (math.pi / F)) / F)
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -100000000.0) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l * Float64(pi / F)) / F));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -100000000.0) || ~(((pi * l) <= 1000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l * (pi / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -100000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e8 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in l around 0 45.8%

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

      1. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -1e8 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 88.0%

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

      1. associate-*l/ 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. associate-/r* 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in l around 0 97.2%

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

      1. associate-/l* 97.1%

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

   6. Simplified 97.1%

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

      1. clear-num 97.1%

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

      2. associate-/r/ 97.1%

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

      3. clear-num 97.2%

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

   8. Applied egg-rr 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000000 \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \]

Alternative 4: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= l -44000000.0) (not (<= l 45000000000000.0)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * (l / (F * F)));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (l <= -44000000.0) or not (l <= 45000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((l <= -44000000.0) || !(l <= 45000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -44000000.0) || ~((l <= 45000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[l, -44000000.0], N[Not[LessEqual[l, 45000000000000.0]], $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 l < -4.4e7 or 4.5e13 < l

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in l around 0 45.8%

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

      1. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -4.4e7 < l < 4.5e13

   1. Initial program 88.0%

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

   2. Taylor expanded in l around 0 87.9%

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

      1. associate-/l* 87.8%

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

      2. associate-/r/ 87.9%

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

      3. unpow2 87.9%

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

   4. Simplified 87.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 5: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= l -44000000.0) (not (<= l 45000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (* PI l) (* F F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) * l) / (F * F));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI * l) / (F * F));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (l <= -44000000.0) or not (l <= 45000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi * l) / (F * F))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((l <= -44000000.0) || !(l <= 45000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * l) / Float64(F * F)));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -44000000.0) || ~((l <= 45000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi * l) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[l, -44000000.0], N[Not[LessEqual[l, 45000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * l), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.4e7 or 4.5e13 < l

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in l around 0 45.8%

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

      1. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -4.4e7 < l < 4.5e13

   1. Initial program 88.0%

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

      1. associate-*l/ 89.7%

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

      2. *-lft-identity 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in l around 0 87.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \ell}{F \cdot F}\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= l -44000000.0) (not (<= l 45000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) * (l / F)) / F);
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI * (l / F)) / F);
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (l <= -44000000.0) or not (l <= 45000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi * (l / F)) / F)
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((l <= -44000000.0) || !(l <= 45000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * Float64(l / F)) / F));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -44000000.0) || ~((l <= 45000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi * (l / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[l, -44000000.0], N[Not[LessEqual[l, 45000000000000.0]], $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 l < -4.4e7 or 4.5e13 < l

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in l around 0 45.8%

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

      1. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -4.4e7 < l < 4.5e13

   1. Initial program 88.0%

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

      1. associate-*l/ 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. associate-/r* 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in l around 0 97.2%

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

      1. associate-/l* 97.1%

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

      2. associate-/r/ 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 7: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= l -44000000.0) (not (<= l 45000000000000.0)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((l <= -44000000.0) || !(l <= 45000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (l <= -44000000.0) or not (l <= 45000000000000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((l <= -44000000.0) || !(l <= 45000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -44000000.0) || ~((l <= 45000000000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[l, -44000000.0], N[Not[LessEqual[l, 45000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.4e7 or 4.5e13 < l

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in l around 0 45.8%

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

      1. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -4.4e7 < l < 4.5e13

   1. Initial program 88.0%

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

      1. associate-*l/ 89.7%

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

      2. *-lft-identity 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in l around 0 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

      1. sub-neg 86.3%

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

      2. distribute-lft-in 86.3%

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

      3. *-commutative 86.3%

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

      4. div-inv 86.3%

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

      5. pow2 86.3%

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

      6. pow-flip 86.3%

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

      7. metadata-eval 86.3%

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

      8. distribute-rgt-neg-in 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. *-commutative 86.3%

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

      2. distribute-lft-in 86.3%

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

      3. distribute-rgt-neg-out 86.3%

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

      4. sub-neg 86.3%

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

      5. *-rgt-identity 86.3%

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

      6. distribute-lft-out-- 86.3%

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

   10. Simplified 86.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -44000000 \lor \neg \left(\ell \leq 45000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 8: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-257} \lor \neg \left(F \cdot F \leq 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F \cdot F} \cdot \left(-\pi\right)\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2.5e-299)
   (* (/ PI F) (/ (- l) F))
   (if (or (<= (* F F) 1e-257) (not (<= (* F F) 1e-205)))
     (* PI l)
     (* (/ l (* F F)) (- PI)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2.5e-299) {
		tmp = (((double) M_PI) / F) * (-l / F);
	} else if (((F * F) <= 1e-257) || !((F * F) <= 1e-205)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l / (F * F)) * -((double) M_PI);
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2.5e-299) {
		tmp = (Math.PI / F) * (-l / F);
	} else if (((F * F) <= 1e-257) || !((F * F) <= 1e-205)) {
		tmp = Math.PI * l;
	} else {
		tmp = (l / (F * F)) * -Math.PI;
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (F * F) <= 2.5e-299:
		tmp = (math.pi / F) * (-l / F)
	elif ((F * F) <= 1e-257) or not ((F * F) <= 1e-205):
		tmp = math.pi * l
	else:
		tmp = (l / (F * F)) * -math.pi
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2.5e-299)
		tmp = Float64(Float64(pi / F) * Float64(Float64(-l) / F));
	elseif ((Float64(F * F) <= 1e-257) || !(Float64(F * F) <= 1e-205))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l / Float64(F * F)) * Float64(-pi));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2.5e-299)
		tmp = (pi / F) * (-l / F);
	elseif (((F * F) <= 1e-257) || ~(((F * F) <= 1e-205)))
		tmp = pi * l;
	else
		tmp = (l / (F * F)) * -pi;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2.5e-299], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 1e-257], N[Not[LessEqual[N[(F * F), $MachinePrecision], 1e-205]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision] * (-Pi)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2.49999999999999978e-299

   1. Initial program 40.0%

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

      1. associate-*l/ 43.0%

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

      2. *-lft-identity 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in l around 0 37.7%

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

      1. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in F around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. associate-/l* 40.5%

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

      3. unpow2 40.5%

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

   9. Simplified 40.5%

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

       1. associate-/l* 40.5%

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

       2. associate-/r/ 58.2%

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

   11. Applied egg-rr 58.2%

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

   if 2.49999999999999978e-299 < (*.f64 F F) < 9.9999999999999998e-258 or 1e-205 < (*.f64 F F)

   1. Initial program 90.9%

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

      1. associate-*l/ 90.9%

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

      2. *-lft-identity 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in l around 0 79.4%

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

      1. unpow2 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in F around inf 86.9%

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

   if 9.9999999999999998e-258 < (*.f64 F F) < 1e-205

   1. Initial program 72.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 70.6%

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

      1. unpow2 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in F around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. associate-/l* 70.9%

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

      3. unpow2 70.9%

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

   9. Simplified 70.9%

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

       1. associate-/r/ 70.7%

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

   11. Applied egg-rr 70.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-257} \lor \neg \left(F \cdot F \leq 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F \cdot F} \cdot \left(-\pi\right)\\ \end{array} \]

Alternative 9: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-257} \lor \neg \left(F \cdot F \leq 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2.5e-299)
   (* (/ PI F) (/ (- l) F))
   (if (or (<= (* F F) 1e-257) (not (<= (* F F) 1e-205)))
     (* PI l)
     (/ (- l) (/ (* F F) PI)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2.5e-299) {
		tmp = (((double) M_PI) / F) * (-l / F);
	} else if (((F * F) <= 1e-257) || !((F * F) <= 1e-205)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = -l / ((F * F) / ((double) M_PI));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2.5e-299) {
		tmp = (Math.PI / F) * (-l / F);
	} else if (((F * F) <= 1e-257) || !((F * F) <= 1e-205)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / ((F * F) / Math.PI);
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (F * F) <= 2.5e-299:
		tmp = (math.pi / F) * (-l / F)
	elif ((F * F) <= 1e-257) or not ((F * F) <= 1e-205):
		tmp = math.pi * l
	else:
		tmp = -l / ((F * F) / math.pi)
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2.5e-299)
		tmp = Float64(Float64(pi / F) * Float64(Float64(-l) / F));
	elseif ((Float64(F * F) <= 1e-257) || !(Float64(F * F) <= 1e-205))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2.5e-299)
		tmp = (pi / F) * (-l / F);
	elseif (((F * F) <= 1e-257) || ~(((F * F) <= 1e-205)))
		tmp = pi * l;
	else
		tmp = -l / ((F * F) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2.5e-299], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 1e-257], N[Not[LessEqual[N[(F * F), $MachinePrecision], 1e-205]], $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) < 2.49999999999999978e-299

   1. Initial program 40.0%

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

      1. associate-*l/ 43.0%

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

      2. *-lft-identity 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in l around 0 37.7%

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

      1. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in F around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. associate-/l* 40.5%

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

      3. unpow2 40.5%

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

   9. Simplified 40.5%

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

       1. associate-/l* 40.5%

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

       2. associate-/r/ 58.2%

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

   11. Applied egg-rr 58.2%

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

   if 2.49999999999999978e-299 < (*.f64 F F) < 9.9999999999999998e-258 or 1e-205 < (*.f64 F F)

   1. Initial program 90.9%

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

      1. associate-*l/ 90.9%

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

      2. *-lft-identity 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in l around 0 79.4%

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

      1. unpow2 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in F around inf 86.9%

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

   if 9.9999999999999998e-258 < (*.f64 F F) < 1e-205

   1. Initial program 72.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 70.6%

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

      1. unpow2 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in F around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. associate-/l* 70.9%

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

      3. unpow2 70.9%

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

   9. Simplified 70.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-257} \lor \neg \left(F \cdot F \leq 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 10: 73.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{-232} \lor \neg \left(F \leq 1.05 \cdot 10^{-161}\right) \land \left(F \leq 4.8 \cdot 10^{-150} \lor \neg \left(F \leq 9.2 \cdot 10^{-128}\right) \land F \leq 1.25 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (or (<= F 2.7e-232)
         (and (not (<= F 1.05e-161))
              (or (<= F 4.8e-150)
                  (and (not (<= F 9.2e-128)) (<= F 1.25e-102)))))
   (* (/ PI F) (/ (- l) F))
   (* PI l)))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((F <= 2.7e-232) || (!(F <= 1.05e-161) && ((F <= 4.8e-150) || (!(F <= 9.2e-128) && (F <= 1.25e-102))))) {
		tmp = (((double) M_PI) / F) * (-l / F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((F <= 2.7e-232) || (!(F <= 1.05e-161) && ((F <= 4.8e-150) || (!(F <= 9.2e-128) && (F <= 1.25e-102))))) {
		tmp = (Math.PI / F) * (-l / F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (F <= 2.7e-232) or (not (F <= 1.05e-161) and ((F <= 4.8e-150) or (not (F <= 9.2e-128) and (F <= 1.25e-102)))):
		tmp = (math.pi / F) * (-l / F)
	else:
		tmp = math.pi * l
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if ((F <= 2.7e-232) || (!(F <= 1.05e-161) && ((F <= 4.8e-150) || (!(F <= 9.2e-128) && (F <= 1.25e-102)))))
		tmp = Float64(Float64(pi / F) * Float64(Float64(-l) / F));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 2.7e-232) || (~((F <= 1.05e-161)) && ((F <= 4.8e-150) || (~((F <= 9.2e-128)) && (F <= 1.25e-102)))))
		tmp = (pi / F) * (-l / F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[Or[LessEqual[F, 2.7e-232], And[N[Not[LessEqual[F, 1.05e-161]], $MachinePrecision], Or[LessEqual[F, 4.8e-150], And[N[Not[LessEqual[F, 9.2e-128]], $MachinePrecision], LessEqual[F, 1.25e-102]]]]], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.6999999999999999e-232 or 1.05e-161 < F < 4.8e-150 or 9.2000000000000003e-128 < F < 1.25000000000000006e-102

   1. Initial program 71.6%

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

      1. associate-*l/ 73.0%

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

      2. *-lft-identity 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in l around 0 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in F around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. associate-/l* 27.9%

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

      3. unpow2 27.9%

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

   9. Simplified 27.9%

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

       1. associate-/l* 27.9%

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

       2. associate-/r/ 35.1%

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

   11. Applied egg-rr 35.1%

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

   if 2.6999999999999999e-232 < F < 1.05e-161 or 4.8e-150 < F < 9.2000000000000003e-128 or 1.25000000000000006e-102 < F

   1. Initial program 82.1%

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

      1. associate-*l/ 82.1%

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

      2. *-lft-identity 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in l around 0 74.3%

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

      1. unpow2 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in F around inf 83.9%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{-232} \lor \neg \left(F \leq 1.05 \cdot 10^{-161}\right) \land \left(F \leq 4.8 \cdot 10^{-150} \lor \neg \left(F \leq 9.2 \cdot 10^{-128}\right) \land F \leq 1.25 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 11: 72.9% accurate, 3.0× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (* PI l))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(Pi * l), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

   1. associate-*l/ 76.7%

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

   2. *-lft-identity 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in l around 0 67.3%

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

   1. unpow2 67.3%

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

6. Simplified 67.3%

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

7. Taylor expanded in F around inf 71.4%

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

8. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))