VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 98.8%

Time: 17.9s

Alternatives: 9

Speedup: 1.5×

• could not determine a ground truth

  1. F = 263903031655.06073
  2. l = -3981970269804643.5

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: 76.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+16) (not (<= (* PI l) 20000.0)))
   (* PI l)
   (+ (* PI l) (/ (* (tan (* PI l)) (/ -1.0 F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+16) || !((((double) M_PI) * l) <= 20000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) * (-1.0 / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+16) || !((Math.PI * l) <= 20000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) * (-1.0 / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+16) or not ((math.pi * l) <= 20000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) * (-1.0 / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+16) || !(Float64(pi * l) <= 20000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) * Float64(-1.0 / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+16) || ~(((pi * l) <= 20000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) * (-1.0 / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 20000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e16 or 2e4 < (*.f64 (PI.f64) l)

   1. Initial program 56.7%

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

      1. associate-*l/ 56.7%

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

      2. *-lft-identity 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e16 < (*.f64 (PI.f64) l) < 2e4

   1. Initial program 91.1%

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

      1. associate-/r* 91.1%

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

      2. metadata-eval 91.1%

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

      3. add-sqr-sqrt 46.9%

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

      4. sqrt-prod 68.6%

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

      5. sqrt-div 68.6%

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

      6. associate-*l/ 68.6%

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

      7. sqrt-div 68.8%

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

      8. metadata-eval 68.8%

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

      9. sqrt-prod 51.5%

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

      10. add-sqr-sqrt 98.9%

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

   3. Applied egg-rr 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+16) (not (<= (* PI l) 20000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+16) || !((((double) M_PI) * l) <= 20000.0)) {
		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) <= -2e+16) || !((Math.PI * l) <= 20000.0)) {
		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) <= -2e+16) or not ((math.pi * l) <= 20000.0):
		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) <= -2e+16) || !(Float64(pi * l) <= 20000.0))
		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) <= -2e+16) || ~(((pi * l) <= 20000.0)))
		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], -2e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 20000.0]], $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) < -2e16 or 2e4 < (*.f64 (PI.f64) l)

   1. Initial program 56.7%

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

      1. associate-*l/ 56.7%

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

      2. *-lft-identity 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in l around 0 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e16 < (*.f64 (PI.f64) l) < 2e4

   1. Initial program 91.1%

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

      1. associate-*l/ 92.0%

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

      2. *-un-lft-identity 92.0%

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

      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%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200 \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -200.0) (not (<= (* PI l) 20000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -200.0) || !((((double) M_PI) * l) <= 20000.0)) {
		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) <= -200.0) || !((Math.PI * l) <= 20000.0)) {
		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) <= -200.0) or not ((math.pi * l) <= 20000.0):
		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) <= -200.0) || !(Float64(pi * l) <= 20000.0))
		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) <= -200.0) || ~(((pi * l) <= 20000.0)))
		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], -200.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 20000.0]], $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) < -200 or 2e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.4%

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

      1. associate-*l/ 57.4%

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

      2. *-lft-identity 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in l around 0 42.2%

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

      1. unpow2 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in F around inf 98.3%

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

   if -200 < (*.f64 (PI.f64) l) < 2e4

   1. Initial program 91.2%

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

      1. associate-*l/ 92.1%

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

      2. *-lft-identity 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in l around 0 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in F around 0 90.1%

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

      1. fma-def 90.1%

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

      2. mul-1-neg 90.1%

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

      3. fma-neg 90.1%

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

      4. *-commutative 90.1%

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

      5. unpow2 90.1%

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

      6. times-frac 97.2%

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

   9. Simplified 97.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200 \lor \neg \left(\pi \cdot \ell \leq 20000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 4: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 180000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 180000.0)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 180000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 180000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 180000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 180000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 180000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 180000.0]], $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 < -0.5 or 1.8e5 < l

   1. Initial program 57.4%

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

      1. associate-*l/ 57.4%

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

      2. *-lft-identity 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in l around 0 42.2%

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

      1. unpow2 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in F around inf 98.3%

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

   if -0.5 < l < 1.8e5

   1. Initial program 91.2%

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

      1. associate-*l/ 92.1%

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

      2. *-lft-identity 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in l around 0 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

      1. sub-neg 89.1%

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

      2. distribute-lft-in 89.1%

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

      3. *-commutative 89.1%

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

      4. div-inv 89.1%

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

      5. pow2 89.1%

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

      6. pow-flip 89.1%

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

      7. metadata-eval 89.1%

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

      8. distribute-rgt-neg-in 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. *-commutative 89.1%

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

      2. distribute-lft-in 89.1%

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

      3. *-commutative 89.1%

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

      4. cancel-sign-sub-inv 89.1%

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

      5. *-lft-identity 89.1%

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

      6. distribute-rgt-out-- 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 180000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 180000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 180000.0)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 180000.0)) {
		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 ((l <= -0.5) || !(l <= 180000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 180000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 180000.0))
		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 ((l <= -0.5) || ~((l <= 180000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 180000.0]], $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 l < -0.5 or 1.8e5 < l

   1. Initial program 57.4%

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

      1. associate-*l/ 57.4%

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

      2. *-lft-identity 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in l around 0 42.2%

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

      1. unpow2 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in F around inf 98.3%

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

   if -0.5 < l < 1.8e5

   1. Initial program 91.2%

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

      1. associate-*l/ 92.1%

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

      2. *-lft-identity 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in l around 0 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 180000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 6: 73.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-290}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 1e-290)
   (/ (* l F) (/ F PI))
   (if (<= (* F F) 2e-277) (* l (/ (- PI) (* F F))) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 1e-290) {
		tmp = (l * F) / (F / ((double) M_PI));
	} else if ((F * F) <= 2e-277) {
		tmp = l * (-((double) M_PI) / (F * F));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 1e-290) {
		tmp = (l * F) / (F / Math.PI);
	} else if ((F * F) <= 2e-277) {
		tmp = l * (-Math.PI / (F * F));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 1e-290:
		tmp = (l * F) / (F / math.pi)
	elif (F * F) <= 2e-277:
		tmp = l * (-math.pi / (F * F))
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 1e-290)
		tmp = Float64(Float64(l * F) / Float64(F / pi));
	elseif (Float64(F * F) <= 2e-277)
		tmp = Float64(l * Float64(Float64(-pi) / Float64(F * F)));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 1e-290)
		tmp = (l * F) / (F / pi);
	elseif ((F * F) <= 2e-277)
		tmp = l * (-pi / (F * F));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 1e-290], N[(N[(l * F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 2e-277], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 1.0000000000000001e-290

   1. Initial program 28.9%

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

      1. associate-*l/ 30.4%

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

      2. *-lft-identity 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in l around 0 24.2%

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

      1. unpow2 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in F around 0 24.2%

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

      1. associate-*r/ 24.2%

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

      2. neg-mul-1 24.2%

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

      3. unpow2 24.2%

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

   9. Simplified 24.2%

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

       1. clear-num 24.2%

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

       2. un-div-inv 25.7%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 7.5%

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

       5. sqr-neg 7.5%

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

       6. sqrt-unprod 7.5%

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

       7. add-sqr-sqrt 7.5%

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

       8. associate-/l* 7.5%

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

       9. associate-/l/ 7.6%

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

       10. div-inv 7.6%

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

       11. associate-/l* 7.6%

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

       12. inv-pow 7.6%

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

       13. metadata-eval 7.6%

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

       14. sqrt-pow1 13.4%

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

       15. exp-to-pow 2.9%

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

       16. associate-*l/ 2.9%

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

   11. Applied egg-rr 59.8%

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

   if 1.0000000000000001e-290 < (*.f64 F F) < 1.99999999999999994e-277

   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/ 88.4%

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

      2. *-lft-identity 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in l around 0 87.1%

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

      1. unpow2 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in F around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. neg-mul-1 87.1%

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

      3. unpow2 87.1%

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

   9. Simplified 87.1%

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

   if 1.99999999999999994e-277 < (*.f64 F F)

   1. Initial program 88.9%

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

      1. associate-*l/ 88.9%

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

      2. *-lft-identity 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in l around 0 77.6%

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

      1. unpow2 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in F around inf 86.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-290}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7: 73.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-290}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 1e-290)
   (/ (* l F) (/ F PI))
   (if (<= (* F F) 2e-277) (* (/ l F) (/ PI (- F))) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 1e-290) {
		tmp = (l * F) / (F / ((double) M_PI));
	} else if ((F * F) <= 2e-277) {
		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 * F) <= 1e-290) {
		tmp = (l * F) / (F / Math.PI);
	} else if ((F * F) <= 2e-277) {
		tmp = (l / F) * (Math.PI / -F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 1e-290:
		tmp = (l * F) / (F / math.pi)
	elif (F * F) <= 2e-277:
		tmp = (l / F) * (math.pi / -F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 1e-290)
		tmp = Float64(Float64(l * F) / Float64(F / pi));
	elseif (Float64(F * F) <= 2e-277)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 1e-290)
		tmp = (l * F) / (F / pi);
	elseif ((F * F) <= 2e-277)
		tmp = (l / F) * (pi / -F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 1e-290], N[(N[(l * F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 2e-277], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 1.0000000000000001e-290

   1. Initial program 28.9%

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

      1. associate-*l/ 30.4%

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

      2. *-lft-identity 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in l around 0 24.2%

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

      1. unpow2 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in F around 0 24.2%

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

      1. associate-*r/ 24.2%

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

      2. neg-mul-1 24.2%

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

      3. unpow2 24.2%

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

   9. Simplified 24.2%

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

       1. clear-num 24.2%

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

       2. un-div-inv 25.7%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 7.5%

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

       5. sqr-neg 7.5%

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

       6. sqrt-unprod 7.5%

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

       7. add-sqr-sqrt 7.5%

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

       8. associate-/l* 7.5%

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

       9. associate-/l/ 7.6%

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

       10. div-inv 7.6%

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

       11. associate-/l* 7.6%

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

       12. inv-pow 7.6%

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

       13. metadata-eval 7.6%

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

       14. sqrt-pow1 13.4%

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

       15. exp-to-pow 2.9%

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

       16. associate-*l/ 2.9%

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

   11. Applied egg-rr 59.8%

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

   if 1.0000000000000001e-290 < (*.f64 F F) < 1.99999999999999994e-277

   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/ 88.4%

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

      2. *-lft-identity 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in l around 0 87.1%

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

      1. unpow2 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in F around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. neg-mul-1 87.1%

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

      3. unpow2 87.1%

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

   9. Simplified 87.1%

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

       1. frac-2neg 87.1%

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

       2. remove-double-neg 87.1%

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

       3. associate-*r/ 87.5%

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

       4. distribute-rgt-neg-in 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. times-frac 87.3%

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

   13. Simplified 87.3%

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

   if 1.99999999999999994e-277 < (*.f64 F F)

   1. Initial program 88.9%

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

      1. associate-*l/ 88.9%

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

      2. *-lft-identity 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in l around 0 77.6%

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

      1. unpow2 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in F around inf 86.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-290}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\pi \cdot \ell}{-F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.2e-145)
   (/ (* l F) (/ F PI))
   (if (<= F 6e-139) (/ (* PI l) (- (* F F))) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.2e-145) {
		tmp = (l * F) / (F / ((double) M_PI));
	} else if (F <= 6e-139) {
		tmp = (((double) M_PI) * l) / -(F * F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 1.2e-145) {
		tmp = (l * F) / (F / Math.PI);
	} else if (F <= 6e-139) {
		tmp = (Math.PI * l) / -(F * F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.2e-145:
		tmp = (l * F) / (F / math.pi)
	elif F <= 6e-139:
		tmp = (math.pi * l) / -(F * F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.2e-145)
		tmp = Float64(Float64(l * F) / Float64(F / pi));
	elseif (F <= 6e-139)
		tmp = Float64(Float64(pi * l) / Float64(-Float64(F * F)));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.2e-145)
		tmp = (l * F) / (F / pi);
	elseif (F <= 6e-139)
		tmp = (pi * l) / -(F * F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.2e-145], N[(N[(l * F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e-139], N[(N[(Pi * l), $MachinePrecision] / (-N[(F * F), $MachinePrecision])), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.20000000000000008e-145

   1. Initial program 64.2%

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

      1. associate-*l/ 64.8%

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

      2. *-lft-identity 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in l around 0 54.5%

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

      1. unpow2 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in F around 0 19.7%

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

      1. associate-*r/ 19.7%

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

      2. neg-mul-1 19.7%

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

      3. unpow2 19.7%

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

   9. Simplified 19.7%

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

       1. clear-num 19.7%

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

       2. un-div-inv 20.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 5.7%

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

       5. sqr-neg 5.7%

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

       6. sqrt-unprod 5.7%

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

       7. add-sqr-sqrt 5.7%

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

       8. associate-/l* 5.7%

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

       9. associate-/l/ 5.8%

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

       10. div-inv 5.8%

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

       11. associate-/l* 5.8%

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

       12. inv-pow 5.8%

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

       13. metadata-eval 5.8%

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

       14. sqrt-pow1 15.4%

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

       15. exp-to-pow 1.2%

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

       16. associate-*l/ 1.2%

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

   11. Applied egg-rr 61.8%

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

   if 1.20000000000000008e-145 < F < 5.9999999999999998e-139

   1. Initial program 81.1%

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

      1. associate-*l/ 81.4%

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

      2. *-lft-identity 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in l around 0 79.7%

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

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in F around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

      3. unpow2 79.7%

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

   9. Simplified 79.7%

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

       1. frac-2neg 79.7%

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

       2. remove-double-neg 79.7%

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

       3. associate-*r/ 80.0%

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

       4. distribute-rgt-neg-in 80.0%

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

   11. Applied egg-rr 80.0%

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

   if 5.9999999999999998e-139 < 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 82.1%

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

      1. unpow2 82.1%

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

   6. Simplified 82.1%

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

   7. Taylor expanded in F around inf 88.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\ell \cdot F}{\frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\pi \cdot \ell}{-F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 9: 74.0% 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 72.7%

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

   1. associate-*l/ 73.1%

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

   2. *-lft-identity 73.1%

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

3. Simplified 73.1%

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

4. Taylor expanded in l around 0 63.5%

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

   1. unpow2 63.5%

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

6. Simplified 63.5%

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

7. Taylor expanded in F around inf 76.9%

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

8. Final simplification 76.9%

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

Reproduce

herbie shell --seed 2023213 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))