VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.1%

Time: 18.3s

Alternatives: 13

Speedup: 1.5×

• could not determine a ground truth

  1. F = -9.719003202571631e+176
  2. l = 3060385170696682.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{1}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+17) (not (<= (* PI l) 10000000.0)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (* F (/ 1.0 (tan (* PI l))))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+17) || !((((double) M_PI) * l) <= 10000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F * (1.0 / tan((((double) M_PI) * l)))));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -1e+17) || !((Math.PI * l) <= 10000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F * (1.0 / Math.tan((Math.PI * l)))));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e+17) or not ((math.pi * l) <= 10000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F * (1.0 / math.tan((math.pi * l)))))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+17) || !(Float64(pi * l) <= 10000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F * Float64(1.0 / tan(Float64(pi * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e+17) || ~(((pi * l) <= 10000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F * (1.0 / tan((pi * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F * N[(1.0 / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e17 or 1e7 < (*.f64 (PI.f64) l)

   1. Initial program 66.6%

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

   2. Taylor expanded in F around 0 66.6%

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

      1. unpow2 66.6%

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

      2. associate-/r* 66.6%

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

      3. *-rgt-identity 66.6%

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

      4. associate-*r/ 66.6%

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

      5. unpow-1 66.6%

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

      6. unpow-1 66.6%

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

      7. pow-sqr 66.6%

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

      8. metadata-eval 66.6%

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

   4. Simplified 66.6%

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

   5. Taylor expanded in l around inf 99.6%

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

   if -1e17 < (*.f64 (PI.f64) l) < 1e7

   1. Initial program 90.5%

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

      1. associate-/r/ 91.6%

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

      2. associate-/l* 99.5%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 51.2%

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

      5. sqrt-prod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 19.1%

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

      8. div-inv 19.1%

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

      9. metadata-eval 19.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 44.3%

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

      12. sqrt-div 43.4%

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

      13. add-sqr-sqrt 71.5%

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

      14. associate-*l/ 71.5%

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

      15. clear-num 71.5%

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

      16. associate-*l/ 71.5%

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

      17. *-un-lft-identity 71.5%

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

   3. Applied egg-rr 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{1}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+17) (not (<= (* PI l) 10000000.0)))
   (* PI l)
   (+ (* PI l) (* (/ (tan (* PI l)) F) (/ -1.0 F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+17) || !((((double) M_PI) * l) <= 10000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) / F) * (-1.0 / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -1e+17) || !((Math.PI * l) <= 10000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) / F) * (-1.0 / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e+17) or not ((math.pi * l) <= 10000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) / F) * (-1.0 / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+17) || !(Float64(pi * l) <= 10000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) / F) * Float64(-1.0 / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e+17) || ~(((pi * l) <= 10000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) / F) * (-1.0 / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e17 or 1e7 < (*.f64 (PI.f64) l)

   1. Initial program 66.6%

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

   2. Taylor expanded in F around 0 66.6%

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

      1. unpow2 66.6%

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

      2. associate-/r* 66.6%

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

      3. *-rgt-identity 66.6%

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

      4. associate-*r/ 66.6%

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

      5. unpow-1 66.6%

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

      6. unpow-1 66.6%

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

      7. pow-sqr 66.6%

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

      8. metadata-eval 66.6%

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

   4. Simplified 66.6%

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

   5. Taylor expanded in l around inf 99.6%

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

   if -1e17 < (*.f64 (PI.f64) l) < 1e7

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 91.6%

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

      3. *-lft-identity 91.6%

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

      4. sqr-neg 91.6%

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

   3. Simplified 91.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+17) (not (<= (* PI l) 10000000.0)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (tan (* PI l)))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+17) || !((((double) M_PI) * l) <= 10000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / tan((((double) M_PI) * l))));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -1e+17) || !((Math.PI * l) <= 10000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / Math.tan((Math.PI * l))));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e+17) or not ((math.pi * l) <= 10000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / math.tan((math.pi * l))))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+17) || !(Float64(pi * l) <= 10000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / tan(Float64(pi * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e+17) || ~(((pi * l) <= 10000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / tan((pi * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e17 or 1e7 < (*.f64 (PI.f64) l)

   1. Initial program 66.6%

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

   2. Taylor expanded in F around 0 66.6%

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

      1. unpow2 66.6%

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

      2. associate-/r* 66.6%

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

      3. *-rgt-identity 66.6%

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

      4. associate-*r/ 66.6%

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

      5. unpow-1 66.6%

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

      6. unpow-1 66.6%

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

      7. pow-sqr 66.6%

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

      8. metadata-eval 66.6%

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

   4. Simplified 66.6%

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

   5. Taylor expanded in l around inf 99.6%

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

   if -1e17 < (*.f64 (PI.f64) l) < 1e7

   1. Initial program 90.5%

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

      1. associate-/r/ 91.6%

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

      2. associate-/l* 99.5%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 51.2%

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

      5. sqrt-prod 66.4%

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

      6. sqr-neg 66.4%

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

      7. sqrt-unprod 19.1%

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

      8. div-inv 19.1%

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

      9. metadata-eval 19.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 44.3%

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

      12. sqrt-div 43.4%

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

      13. add-sqr-sqrt 71.5%

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

      14. associate-*l/ 71.5%

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

      15. clear-num 71.5%

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

      16. associate-*l/ 71.5%

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

      17. *-un-lft-identity 71.5%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 10000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-19}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{1}{\pi \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5.0) (not (<= (* PI l) 4e-19)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (* F (/ 1.0 (* PI l)))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5.0) || !((((double) M_PI) * l) <= 4e-19)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F * (1.0 / (((double) M_PI) * l))));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5.0) || !((Math.PI * l) <= 4e-19)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F * (1.0 / (Math.PI * l))));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5.0) or not ((math.pi * l) <= 4e-19):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F * (1.0 / (math.pi * l))))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5.0) || !(Float64(pi * l) <= 4e-19))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F * Float64(1.0 / Float64(pi * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5.0) || ~(((pi * l) <= 4e-19)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F * (1.0 / (pi * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 4e-19]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F * N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5 or 3.9999999999999999e-19 < (*.f64 (PI.f64) l)

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -5 < (*.f64 (PI.f64) l) < 3.9999999999999999e-19

   1. Initial program 90.1%

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

      1. associate-/r/ 91.2%

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

      2. associate-/l* 99.5%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 51.7%

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

      5. sqrt-prod 65.4%

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

      6. sqr-neg 65.4%

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

      7. sqrt-unprod 17.8%

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

      8. div-inv 17.8%

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

      9. metadata-eval 17.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 43.6%

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

      12. sqrt-div 42.6%

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

      13. add-sqr-sqrt 70.8%

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

      14. associate-*l/ 70.8%

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

      15. clear-num 70.8%

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

      16. associate-*l/ 70.8%

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

      17. *-un-lft-identity 70.8%

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

   3. Applied egg-rr 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in l around 0 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-19}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{1}{\pi \cdot \ell}}\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 5.5e-12)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (* PI l))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / (((double) M_PI) * l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / (Math.PI * l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 5.5e-12):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / (math.pi * l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 5.5e-12))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / Float64(pi * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 5.5e-12)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / (pi * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 5.5e-12]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 5.5000000000000004e-12 < l

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -0.5 < l < 5.5000000000000004e-12

   1. Initial program 90.1%

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

      1. associate-/r/ 91.2%

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

      2. associate-/l* 99.5%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 51.7%

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

      5. sqrt-prod 65.4%

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

      6. sqr-neg 65.4%

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

      7. sqrt-unprod 17.8%

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

      8. div-inv 17.8%

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

      9. metadata-eval 17.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 43.6%

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

      12. sqrt-div 42.6%

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

      13. add-sqr-sqrt 70.8%

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

      14. associate-*l/ 70.8%

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

      15. clear-num 70.8%

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

      16. associate-*l/ 70.8%

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

      17. *-un-lft-identity 70.8%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell}}\\ \end{array} \]

Alternative 6: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.58 \cdot 10^{-54}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-136}:\\ \;\;\;\;\ell \cdot \left({F}^{-2} \cdot \left(-\pi\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-240} \lor \neg \left(\ell \leq 2.25 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{{F}^{2}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -1.58e-54)
   (* PI l)
   (if (<= l -2.25e-136)
     (* l (* (pow F -2.0) (- PI)))
     (if (or (<= l 2.2e-240) (not (<= l 2.25e-46)))
       (* PI l)
       (/ (- l) (/ (pow F 2.0) PI))))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -1.58e-54) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -2.25e-136) {
		tmp = l * (pow(F, -2.0) * -((double) M_PI));
	} else if ((l <= 2.2e-240) || !(l <= 2.25e-46)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = -l / (pow(F, 2.0) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -1.58e-54) {
		tmp = Math.PI * l;
	} else if (l <= -2.25e-136) {
		tmp = l * (Math.pow(F, -2.0) * -Math.PI);
	} else if ((l <= 2.2e-240) || !(l <= 2.25e-46)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / (Math.pow(F, 2.0) / Math.PI);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -1.58e-54:
		tmp = math.pi * l
	elif l <= -2.25e-136:
		tmp = l * (math.pow(F, -2.0) * -math.pi)
	elif (l <= 2.2e-240) or not (l <= 2.25e-46):
		tmp = math.pi * l
	else:
		tmp = -l / (math.pow(F, 2.0) / math.pi)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -1.58e-54)
		tmp = Float64(pi * l);
	elseif (l <= -2.25e-136)
		tmp = Float64(l * Float64((F ^ -2.0) * Float64(-pi)));
	elseif ((l <= 2.2e-240) || !(l <= 2.25e-46))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64((F ^ 2.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -1.58e-54)
		tmp = pi * l;
	elseif (l <= -2.25e-136)
		tmp = l * ((F ^ -2.0) * -pi);
	elseif ((l <= 2.2e-240) || ~((l <= 2.25e-46)))
		tmp = pi * l;
	else
		tmp = -l / ((F ^ 2.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -1.58e-54], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -2.25e-136], N[(l * N[(N[Power[F, -2.0], $MachinePrecision] * (-Pi)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.2e-240], N[Not[LessEqual[l, 2.25e-46]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(N[Power[F, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.57999999999999989e-54 or -2.24999999999999986e-136 < l < 2.1999999999999999e-240 or 2.25e-46 < l

   1. Initial program 74.6%

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

   2. Taylor expanded in F around 0 74.6%

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

      1. unpow2 74.6%

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

      2. associate-/r* 74.6%

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

      3. *-rgt-identity 74.6%

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

      4. associate-*r/ 74.5%

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

      5. unpow-1 74.5%

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

      6. unpow-1 74.5%

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

      7. pow-sqr 74.6%

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

      8. metadata-eval 74.6%

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

   4. Simplified 74.6%

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

   5. Taylor expanded in l around inf 86.7%

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

   if -1.57999999999999989e-54 < l < -2.24999999999999986e-136

   1. Initial program 95.7%

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

      1. sqr-neg 95.7%

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

      2. associate-*l/ 95.7%

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

      3. *-lft-identity 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in l around 0 95.6%

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

   5. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.7%

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

      3. distribute-neg-frac 70.7%

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

      4. unpow2 70.7%

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

      5. sqr-neg 70.7%

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

      6. associate-*l/ 70.7%

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

      7. distribute-neg-frac 70.7%

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

      8. associate-*l/ 70.7%

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

      9. sqr-neg 70.7%

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

      10. associate-*l/ 70.7%

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

      11. associate-/l/ 74.6%

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

      12. remove-double-neg 74.6%

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

      13. neg-mul-1 74.6%

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

      14. distribute-lft-neg-in 74.6%

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

      15. metadata-eval 74.6%

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

      16. associate-*l/ 74.6%

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

      17. associate-*l/ 70.7%

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

   7. Simplified 70.8%

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

   if 2.1999999999999999e-240 < l < 2.25e-46

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. associate-*l/ 90.9%

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

      3. *-lft-identity 90.9%

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

      4. sqr-neg 90.9%

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

   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 89.1%

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

   5. Taylor expanded in F around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. associate-/l* 55.8%

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

   7. Simplified 55.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.58 \cdot 10^{-54}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-136}:\\ \;\;\;\;\ell \cdot \left({F}^{-2} \cdot \left(-\pi\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-240} \lor \neg \left(\ell \leq 2.25 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{{F}^{2}}{\pi}}\\ \end{array} \]

Alternative 7: 72.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.58 \cdot 10^{-54}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.36 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \left({F}^{-2} \cdot \left(-\pi\right)\right)\\ \mathbf{elif}\;\ell \leq 3.15 \cdot 10^{-251} \lor \neg \left(\ell \leq 2.6 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{{F}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -1.58e-54)
   (* PI l)
   (if (<= l -1.36e-135)
     (* l (* (pow F -2.0) (- PI)))
     (if (or (<= l 3.15e-251) (not (<= l 2.6e-46)))
       (* PI l)
       (* PI (/ (- l) (pow F 2.0)))))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -1.58e-54) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -1.36e-135) {
		tmp = l * (pow(F, -2.0) * -((double) M_PI));
	} else if ((l <= 3.15e-251) || !(l <= 2.6e-46)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (-l / pow(F, 2.0));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -1.58e-54) {
		tmp = Math.PI * l;
	} else if (l <= -1.36e-135) {
		tmp = l * (Math.pow(F, -2.0) * -Math.PI);
	} else if ((l <= 3.15e-251) || !(l <= 2.6e-46)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (-l / Math.pow(F, 2.0));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -1.58e-54:
		tmp = math.pi * l
	elif l <= -1.36e-135:
		tmp = l * (math.pow(F, -2.0) * -math.pi)
	elif (l <= 3.15e-251) or not (l <= 2.6e-46):
		tmp = math.pi * l
	else:
		tmp = math.pi * (-l / math.pow(F, 2.0))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -1.58e-54)
		tmp = Float64(pi * l);
	elseif (l <= -1.36e-135)
		tmp = Float64(l * Float64((F ^ -2.0) * Float64(-pi)));
	elseif ((l <= 3.15e-251) || !(l <= 2.6e-46))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(Float64(-l) / (F ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -1.58e-54)
		tmp = pi * l;
	elseif (l <= -1.36e-135)
		tmp = l * ((F ^ -2.0) * -pi);
	elseif ((l <= 3.15e-251) || ~((l <= 2.6e-46)))
		tmp = pi * l;
	else
		tmp = pi * (-l / (F ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -1.58e-54], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -1.36e-135], N[(l * N[(N[Power[F, -2.0], $MachinePrecision] * (-Pi)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 3.15e-251], N[Not[LessEqual[l, 2.6e-46]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[((-l) / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.57999999999999989e-54 or -1.36e-135 < l < 3.1499999999999999e-251 or 2.6000000000000002e-46 < l

   1. Initial program 74.6%

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

   2. Taylor expanded in F around 0 74.6%

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

      1. unpow2 74.6%

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

      2. associate-/r* 74.6%

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

      3. *-rgt-identity 74.6%

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

      4. associate-*r/ 74.5%

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

      5. unpow-1 74.5%

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

      6. unpow-1 74.5%

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

      7. pow-sqr 74.6%

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

      8. metadata-eval 74.6%

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

   4. Simplified 74.6%

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

   5. Taylor expanded in l around inf 86.7%

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

   if -1.57999999999999989e-54 < l < -1.36e-135

   1. Initial program 95.7%

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

      1. sqr-neg 95.7%

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

      2. associate-*l/ 95.7%

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

      3. *-lft-identity 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in l around 0 95.6%

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

   5. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.7%

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

      3. distribute-neg-frac 70.7%

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

      4. unpow2 70.7%

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

      5. sqr-neg 70.7%

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

      6. associate-*l/ 70.7%

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

      7. distribute-neg-frac 70.7%

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

      8. associate-*l/ 70.7%

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

      9. sqr-neg 70.7%

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

      10. associate-*l/ 70.7%

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

      11. associate-/l/ 74.6%

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

      12. remove-double-neg 74.6%

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

      13. neg-mul-1 74.6%

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

      14. distribute-lft-neg-in 74.6%

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

      15. metadata-eval 74.6%

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

      16. associate-*l/ 74.6%

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

      17. associate-*l/ 70.7%

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

   7. Simplified 70.8%

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

   if 3.1499999999999999e-251 < l < 2.6000000000000002e-46

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. associate-*l/ 90.9%

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

      3. *-lft-identity 90.9%

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

      4. sqr-neg 90.9%

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

   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 90.9%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/l* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.5%

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

      2. div-inv 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in F around 0 55.8%

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

       1. mul-1-neg 55.8%

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

       2. associate-/l* 55.8%

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

       3. distribute-frac-neg 55.8%

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

       4. associate-/r/ 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.58 \cdot 10^{-54}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.36 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \left({F}^{-2} \cdot \left(-\pi\right)\right)\\ \mathbf{elif}\;\ell \leq 3.15 \cdot 10^{-251} \lor \neg \left(\ell \leq 2.6 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{{F}^{2}}\\ \end{array} \]

Alternative 8: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 5.5e-12)))
   (* PI l)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / F) * (((double) M_PI) / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 5.5e-12):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / F) * (math.pi / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 5.5e-12))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / F) * Float64(pi / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 5.5e-12)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) * (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 5.5e-12]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 5.5000000000000004e-12 < l

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -0.5 < l < 5.5000000000000004e-12

   1. Initial program 90.1%

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

      1. sqr-neg 90.1%

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

      2. associate-*l/ 91.2%

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

      3. *-lft-identity 91.2%

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

      4. sqr-neg 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in l around 0 91.2%

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

      1. *-commutative 91.2%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 9: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 5.5e-12)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) * (l / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI * (l / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 5.5e-12):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi * (l / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 5.5e-12))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * Float64(l / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 5.5e-12)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi * (l / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 5.5e-12]], $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 < -0.5 or 5.5000000000000004e-12 < l

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -0.5 < l < 5.5000000000000004e-12

   1. Initial program 90.1%

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

      1. sqr-neg 90.1%

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

      2. associate-*l/ 91.2%

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

      3. *-lft-identity 91.2%

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

      4. sqr-neg 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in l around 0 91.2%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.5%

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

      2. associate-/l* 99.6%

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

      3. *-commutative 99.6%

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

      4. div-inv 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 10: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 5.5e-12)))
   (* PI l)
   (- (* PI l) (/ (/ l F) (/ F PI)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / F) / (F / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) / (F / Math.PI));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.5) or not (l <= 5.5e-12):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / F) / (F / math.pi))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.5) || !(l <= 5.5e-12))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / F) / Float64(F / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((l <= 5.5e-12)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) / (F / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.5], N[Not[LessEqual[l, 5.5e-12]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 5.5000000000000004e-12 < l

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -0.5 < l < 5.5000000000000004e-12

   1. Initial program 90.1%

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

      1. sqr-neg 90.1%

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

      2. associate-*l/ 91.2%

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

      3. *-lft-identity 91.2%

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

      4. sqr-neg 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in l around 0 91.2%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.5%

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

      2. div-inv 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \end{array} \]

Alternative 11: 46.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-259} \lor \neg \left(F \leq 5.5 \cdot 10^{-46}\right) \land F \leq 8.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\ell}{\frac{{F}^{2}}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 2.4e-259) (and (not (<= F 5.5e-46)) (<= F 8.6e-5)))
   (/ (- l) (/ (pow F 2.0) PI))
   (* PI l)))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 2.4e-259) || (!(F <= 5.5e-46) && (F <= 8.6e-5))) {
		tmp = -l / (pow(F, 2.0) / ((double) M_PI));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 2.4e-259) || (!(F <= 5.5e-46) && (F <= 8.6e-5))) {
		tmp = -l / (Math.pow(F, 2.0) / Math.PI);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 2.4e-259) or (not (F <= 5.5e-46) and (F <= 8.6e-5)):
		tmp = -l / (math.pow(F, 2.0) / math.pi)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 2.4e-259) || (!(F <= 5.5e-46) && (F <= 8.6e-5)))
		tmp = Float64(Float64(-l) / Float64((F ^ 2.0) / pi));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 2.4e-259) || (~((F <= 5.5e-46)) && (F <= 8.6e-5)))
		tmp = -l / ((F ^ 2.0) / pi);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 2.4e-259], And[N[Not[LessEqual[F, 5.5e-46]], $MachinePrecision], LessEqual[F, 8.6e-5]]], N[((-l) / N[(N[Power[F, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.4000000000000001e-259 or 5.49999999999999983e-46 < F < 8.6000000000000003e-5

   1. Initial program 79.2%

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

      1. sqr-neg 79.2%

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

      2. associate-*l/ 80.0%

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

      3. *-lft-identity 80.0%

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

      4. sqr-neg 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in l around 0 67.9%

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

   5. Taylor expanded in F around 0 31.7%

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

      1. mul-1-neg 31.7%

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

      2. associate-/l* 31.7%

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

   7. Simplified 31.7%

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

   if 2.4000000000000001e-259 < F < 5.49999999999999983e-46 or 8.6000000000000003e-5 < F

   1. Initial program 78.6%

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

   2. Taylor expanded in F around 0 78.6%

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

      1. unpow2 78.6%

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

      2. associate-/r* 78.6%

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

      3. *-rgt-identity 78.6%

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

      4. associate-*r/ 78.6%

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

      5. unpow-1 78.6%

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

      6. unpow-1 78.6%

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

      7. pow-sqr 78.7%

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

      8. metadata-eval 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in l around inf 83.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-259} \lor \neg \left(F \leq 5.5 \cdot 10^{-46}\right) \land F \leq 8.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\ell}{\frac{{F}^{2}}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 12: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.5) (not (<= l 5.5e-12)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(l <= 5.5e-12)) {
		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 <= 5.5e-12)) {
		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 <= 5.5e-12):
		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 <= 5.5e-12))
		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 <= 5.5e-12)))
		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, 5.5e-12]], $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 5.5000000000000004e-12 < l

   1. Initial program 68.4%

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

   2. Taylor expanded in F around 0 68.4%

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

      1. unpow2 68.4%

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

      2. associate-/r* 68.4%

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

      3. *-rgt-identity 68.4%

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

      4. associate-*r/ 68.4%

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

      5. unpow-1 68.4%

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

      6. unpow-1 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in l around inf 98.1%

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

   if -0.5 < l < 5.5000000000000004e-12

   1. Initial program 90.1%

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

      1. sqr-neg 90.1%

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

      2. associate-*l/ 91.2%

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

      3. *-lft-identity 91.2%

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

      4. sqr-neg 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in l around 0 90.0%

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

   5. Taylor expanded in F around 0 91.2%

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

   7. Simplified 90.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.5 \lor \neg \left(\ell \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 13: 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 79.0%

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

2. Taylor expanded in F around 0 79.0%

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

   1. unpow2 79.0%

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

   2. associate-/r* 79.0%

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

   3. *-rgt-identity 79.0%

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

   4. associate-*r/ 78.9%

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

   5. unpow-1 78.9%

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

   6. unpow-1 78.9%

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

   7. pow-sqr 79.0%

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

   8. metadata-eval 79.0%

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

4. Simplified 79.0%

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

5. Taylor expanded in l around inf 72.8%

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

6. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2023300 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))