VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 98.9%

Time: 13.6s

Alternatives: 14

Speedup: 1.4×

• could not determine a ground truth

  1. F = 4.48267107910094e-218
  2. l = 2012372594190702.5

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: 76.2% 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: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+19) (not (<= (* PI l) 2000.0)))
   (* 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) <= -2e+19) || !((((double) M_PI) * l) <= 2000.0)) {
		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) <= -2e+19) || !((Math.PI * l) <= 2000.0)) {
		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) <= -2e+19) or not ((math.pi * l) <= 2000.0):
		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) <= -2e+19) || !(Float64(pi * l) <= 2000.0))
		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) <= -2e+19) || ~(((pi * l) <= 2000.0)))
		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], -2e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2000.0]], $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) < -2e19 or 2e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.0%

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

      1. associate-*l/ 71.0%

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

      2. *-lft-identity 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e19 < (*.f64 (PI.f64) l) < 2e3

   1. Initial program 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

   3. Simplified 86.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternative 2: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 0.4:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -5.0)
   (* PI l)
   (if (<= (* PI l) -5e-290)
     (* (* PI l) (- 1.0 (pow F -2.0)))
     (if (<= (* PI l) 2e-279)
       (* (/ l F) (/ PI (- F)))
       (if (<= (* PI l) 0.4) (- (* PI l) (* PI (/ l (* F F)))) (* PI l))))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -5.0) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= -5e-290) {
		tmp = (((double) M_PI) * l) * (1.0 - pow(F, -2.0));
	} else if ((((double) M_PI) * l) <= 2e-279) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if ((((double) M_PI) * l) <= 0.4) {
		tmp = (((double) M_PI) * l) - (((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 ((Math.PI * l) <= -5.0) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= -5e-290) {
		tmp = (Math.PI * l) * (1.0 - Math.pow(F, -2.0));
	} else if ((Math.PI * l) <= 2e-279) {
		tmp = (l / F) * (Math.PI / -F);
	} else if ((Math.PI * l) <= 0.4) {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (math.pi * l) <= -5.0:
		tmp = math.pi * l
	elif (math.pi * l) <= -5e-290:
		tmp = (math.pi * l) * (1.0 - math.pow(F, -2.0))
	elif (math.pi * l) <= 2e-279:
		tmp = (l / F) * (math.pi / -F)
	elif (math.pi * l) <= 0.4:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= -5.0)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= -5e-290)
		tmp = Float64(Float64(pi * l) * Float64(1.0 - (F ^ -2.0)));
	elseif (Float64(pi * l) <= 2e-279)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif (Float64(pi * l) <= 0.4)
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((pi * l) <= -5.0)
		tmp = pi * l;
	elseif ((pi * l) <= -5e-290)
		tmp = (pi * l) * (1.0 - (F ^ -2.0));
	elseif ((pi * l) <= 2e-279)
		tmp = (l / F) * (pi / -F);
	elseif ((pi * l) <= 0.4)
		tmp = (pi * l) - (pi * (l / (F * F)));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], -5.0], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], -5e-290], N[(N[(Pi * l), $MachinePrecision] * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 2e-279], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 0.4], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (PI.f64) l) < -5 or 0.40000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -5 < (*.f64 (PI.f64) l) < -5.0000000000000001e-290

   1. Initial program 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

      1. add-sqr-sqrt 35.4%

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

      2. pow2 35.4%

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

      3. sub-neg 35.4%

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

      4. sub-neg 35.4%

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

      5. div-inv 35.4%

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

      6. pow2 35.4%

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

      7. pow-flip 35.4%

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

      8. metadata-eval 35.4%

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

   8. Applied egg-rr 35.4%

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

      1. unpow2 35.4%

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

      2. add-sqr-sqrt 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. *-un-lft-identity 88.6%

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

      6. distribute-rgt-out-- 88.6%

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

      7. *-commutative 88.6%

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

      8. associate-*l* 88.6%

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

   10. Applied egg-rr 88.6%

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

   if -5.0000000000000001e-290 < (*.f64 (PI.f64) l) < 2.00000000000000011e-279

   1. Initial program 46.0%

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

      1. associate-*l/ 46.0%

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

      2. *-lft-identity 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in l around 0 46.1%

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

      1. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in F around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. associate-/l* 26.9%

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

      3. unpow2 26.9%

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

      4. associate-*r/ 27.1%

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

      5. distribute-neg-frac 27.1%

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

      6. associate-*r/ 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in l around 0 26.9%

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

       1. unpow2 26.9%

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

       2. associate-*r/ 26.9%

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

       3. *-commutative 26.9%

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

       4. associate-*l/ 26.9%

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

       5. associate-*r* 26.9%

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

       6. associate-/r* 27.0%

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

       7. metadata-eval 27.0%

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

       8. associate-/r* 27.0%

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

       9. neg-mul-1 27.0%

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

       10. associate-/r* 26.9%

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

       11. *-commutative 26.9%

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

       12. associate-/r/ 26.8%

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

       13. associate-*l/ 26.9%

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

       14. *-lft-identity 26.9%

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

       15. associate-/l* 27.0%

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

       16. associate-/r/ 80.4%

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

   12. Simplified 80.4%

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

   if 2.00000000000000011e-279 < (*.f64 (PI.f64) l) < 0.40000000000000002

   1. Initial program 93.1%

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

   2. Taylor expanded in l around 0 92.8%

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

      1. associate-/l* 92.8%

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

      2. associate-/r/ 92.8%

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

      3. unpow2 92.8%

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

   4. Simplified 92.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F \cdot F} \cdot \pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 0.4:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {F}^{-2}\\ \mathbf{if}\;\pi \cdot \ell \leq -5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot t_0\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 0.4:\\ \;\;\;\;\pi \cdot \left(\ell \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow F -2.0))))
   (if (<= (* PI l) -5.0)
     (* PI l)
     (if (<= (* PI l) -5e-290)
       (* (* PI l) t_0)
       (if (<= (* PI l) 2e-279)
         (* (/ l F) (/ PI (- F)))
         (if (<= (* PI l) 0.4) (* PI (* l t_0)) (* PI l)))))))

C

double code(double F, double l) {
	double t_0 = 1.0 - pow(F, -2.0);
	double tmp;
	if ((((double) M_PI) * l) <= -5.0) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= -5e-290) {
		tmp = (((double) M_PI) * l) * t_0;
	} else if ((((double) M_PI) * l) <= 2e-279) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if ((((double) M_PI) * l) <= 0.4) {
		tmp = ((double) M_PI) * (l * t_0);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = 1.0 - Math.pow(F, -2.0);
	double tmp;
	if ((Math.PI * l) <= -5.0) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= -5e-290) {
		tmp = (Math.PI * l) * t_0;
	} else if ((Math.PI * l) <= 2e-279) {
		tmp = (l / F) * (Math.PI / -F);
	} else if ((Math.PI * l) <= 0.4) {
		tmp = Math.PI * (l * t_0);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = 1.0 - math.pow(F, -2.0)
	tmp = 0
	if (math.pi * l) <= -5.0:
		tmp = math.pi * l
	elif (math.pi * l) <= -5e-290:
		tmp = (math.pi * l) * t_0
	elif (math.pi * l) <= 2e-279:
		tmp = (l / F) * (math.pi / -F)
	elif (math.pi * l) <= 0.4:
		tmp = math.pi * (l * t_0)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(1.0 - (F ^ -2.0))
	tmp = 0.0
	if (Float64(pi * l) <= -5.0)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= -5e-290)
		tmp = Float64(Float64(pi * l) * t_0);
	elseif (Float64(pi * l) <= 2e-279)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif (Float64(pi * l) <= 0.4)
		tmp = Float64(pi * Float64(l * t_0));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = 1.0 - (F ^ -2.0);
	tmp = 0.0;
	if ((pi * l) <= -5.0)
		tmp = pi * l;
	elseif ((pi * l) <= -5e-290)
		tmp = (pi * l) * t_0;
	elseif ((pi * l) <= 2e-279)
		tmp = (l / F) * (pi / -F);
	elseif ((pi * l) <= 0.4)
		tmp = pi * (l * t_0);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(Pi * l), $MachinePrecision], -5.0], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], -5e-290], N[(N[(Pi * l), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 2e-279], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 0.4], N[(Pi * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (PI.f64) l) < -5 or 0.40000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -5 < (*.f64 (PI.f64) l) < -5.0000000000000001e-290

   1. Initial program 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

      1. add-sqr-sqrt 35.4%

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

      2. pow2 35.4%

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

      3. sub-neg 35.4%

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

      4. sub-neg 35.4%

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

      5. div-inv 35.4%

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

      6. pow2 35.4%

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

      7. pow-flip 35.4%

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

      8. metadata-eval 35.4%

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

   8. Applied egg-rr 35.4%

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

      1. unpow2 35.4%

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

      2. add-sqr-sqrt 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. *-un-lft-identity 88.6%

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

      6. distribute-rgt-out-- 88.6%

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

      7. *-commutative 88.6%

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

      8. associate-*l* 88.6%

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

   10. Applied egg-rr 88.6%

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

   if -5.0000000000000001e-290 < (*.f64 (PI.f64) l) < 2.00000000000000011e-279

   1. Initial program 46.0%

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

      1. associate-*l/ 46.0%

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

      2. *-lft-identity 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in l around 0 46.1%

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

      1. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in F around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. associate-/l* 26.9%

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

      3. unpow2 26.9%

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

      4. associate-*r/ 27.1%

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

      5. distribute-neg-frac 27.1%

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

      6. associate-*r/ 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in l around 0 26.9%

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

       1. unpow2 26.9%

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

       2. associate-*r/ 26.9%

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

       3. *-commutative 26.9%

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

       4. associate-*l/ 26.9%

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

       5. associate-*r* 26.9%

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

       6. associate-/r* 27.0%

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

       7. metadata-eval 27.0%

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

       8. associate-/r* 27.0%

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

       9. neg-mul-1 27.0%

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

       10. associate-/r* 26.9%

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

       11. *-commutative 26.9%

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

       12. associate-/r/ 26.8%

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

       13. associate-*l/ 26.9%

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

       14. *-lft-identity 26.9%

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

       15. associate-/l* 27.0%

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

       16. associate-/r/ 80.4%

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

   12. Simplified 80.4%

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

   if 2.00000000000000011e-279 < (*.f64 (PI.f64) l) < 0.40000000000000002

   1. Initial program 93.1%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in l around 0 92.8%

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

      1. unpow2 92.8%

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

   6. Simplified 92.8%

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

      1. add-sqr-sqrt 52.1%

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

      2. pow2 52.1%

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

      3. sub-neg 52.1%

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

      4. sub-neg 52.1%

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

      5. div-inv 52.1%

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

      6. pow2 52.1%

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

      7. pow-flip 52.1%

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

      8. metadata-eval 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. unpow2 52.1%

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

      2. add-sqr-sqrt 92.8%

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

      3. *-commutative 92.8%

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

      4. *-un-lft-identity 92.8%

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

      5. distribute-rgt-out-- 92.7%

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

      6. *-commutative 92.7%

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

      7. associate-*r* 92.8%

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

   10. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(1 - {F}^{-2}\right)\right) \cdot \pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 0.4:\\ \;\;\;\;\pi \cdot \left(\ell \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+19) (not (<= (* PI l) 2000.0)))
   (* 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) <= -2e+19) || !((((double) M_PI) * l) <= 2000.0)) {
		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) <= -2e+19) || !((Math.PI * l) <= 2000.0)) {
		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) <= -2e+19) or not ((math.pi * l) <= 2000.0):
		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) <= -2e+19) || !(Float64(pi * l) <= 2000.0))
		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) <= -2e+19) || ~(((pi * l) <= 2000.0)))
		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], -2e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2000.0]], $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) < -2e19 or 2e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.0%

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

      1. associate-*l/ 71.0%

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

      2. *-lft-identity 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e19 < (*.f64 (PI.f64) l) < 2e3

   1. Initial program 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

   3. Simplified 86.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \]

Alternative 5: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+19) (not (<= (* PI l) 2000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+19) || !((((double) M_PI) * l) <= 2000.0)) {
		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+19) || !((Math.PI * l) <= 2000.0)) {
		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+19) or not ((math.pi * l) <= 2000.0):
		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+19) || !(Float64(pi * l) <= 2000.0))
		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+19) || ~(((pi * l) <= 2000.0)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2000.0]], $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) < -2e19 or 2e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.0%

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

      1. associate-*l/ 71.0%

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

      2. *-lft-identity 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e19 < (*.f64 (PI.f64) l) < 2e3

   1. Initial program 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-un-lft-identity 86.2%

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

      3. associate-/r* 99.2%

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

   3. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 2000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 6: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\pi \cdot \ell}{F} \cdot \frac{-1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5.0) (not (<= (* PI l) 0.4)))
   (* PI l)
   (+ (* PI l) (* (/ (* PI l) F) (/ -1.0 F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5.0) || !((((double) M_PI) * l) <= 0.4)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + (((((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) <= -5.0) || !((Math.PI * l) <= 0.4)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + (((Math.PI * l) / F) * (-1.0 / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5.0) or not ((math.pi * l) <= 0.4):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + (((math.pi * l) / F) * (-1.0 / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5.0) || !(Float64(pi * l) <= 0.4))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(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) <= -5.0) || ~(((pi * l) <= 0.4)))
		tmp = pi * l;
	else
		tmp = (pi * l) + (((pi * l) / F) * (-1.0 / F));
	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], 0.4]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5 or 0.40000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -5 < (*.f64 (PI.f64) l) < 0.40000000000000002

   1. Initial program 86.1%

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

      1. associate-*l/ 86.1%

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

      2. *-lft-identity 86.1%

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

   3. Simplified 86.1%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in l around 0 99.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\pi \cdot \ell}{F} \cdot \frac{-1}{F}\\ \end{array} \]

Alternative 7: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5.0) (not (<= (* PI l) 0.4)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (* PI l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5.0) || !((((double) M_PI) * l) <= 0.4)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / (((double) M_PI) * l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5.0) || !((Math.PI * l) <= 0.4)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / (Math.PI * l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5.0) or not ((math.pi * l) <= 0.4):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / (math.pi * l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5.0) || !(Float64(pi * l) <= 0.4))
		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 (((pi * l) <= -5.0) || ~(((pi * l) <= 0.4)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / (pi * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.4]], $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 (*.f64 (PI.f64) l) < -5 or 0.40000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -5 < (*.f64 (PI.f64) l) < 0.40000000000000002

   1. Initial program 86.1%

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

      1. associate-*l/ 86.1%

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

      2. *-lft-identity 86.1%

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

   3. Simplified 86.1%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in l around 0 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell}}\\ \end{array} \]

Alternative 8: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5.0) (not (<= (* PI l) 0.4)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5.0) || !((((double) M_PI) * l) <= 0.4)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((((double) M_PI) * l) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5.0) || !((Math.PI * l) <= 0.4)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (((Math.PI * l) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5.0) or not ((math.pi * l) <= 0.4):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (((math.pi * l) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5.0) || !(Float64(pi * l) <= 0.4))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5.0) || ~(((pi * l) <= 0.4)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (((pi * l) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.4]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5 or 0.40000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -5 < (*.f64 (PI.f64) l) < 0.40000000000000002

   1. Initial program 86.1%

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

      1. associate-*l/ 86.1%

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

      2. *-un-lft-identity 86.1%

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

      3. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in l around 0 99.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \lor \neg \left(\pi \cdot \ell \leq 0.4\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 9: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* l (* PI (- 1.0 (pow F -2.0))))))
   (if (<= l -1.5)
     (* PI l)
     (if (<= l -1.1e-293)
       t_0
       (if (<= l 5.6e-280)
         (* (/ l F) (/ PI (- F)))
         (if (<= l 0.5) t_0 (* PI l)))))))

C

double code(double F, double l) {
	double t_0 = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	double tmp;
	if (l <= -1.5) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -1.1e-293) {
		tmp = t_0;
	} else if (l <= 5.6e-280) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if (l <= 0.5) {
		tmp = t_0;
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	double tmp;
	if (l <= -1.5) {
		tmp = Math.PI * l;
	} else if (l <= -1.1e-293) {
		tmp = t_0;
	} else if (l <= 5.6e-280) {
		tmp = (l / F) * (Math.PI / -F);
	} else if (l <= 0.5) {
		tmp = t_0;
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	tmp = 0
	if l <= -1.5:
		tmp = math.pi * l
	elif l <= -1.1e-293:
		tmp = t_0
	elif l <= 5.6e-280:
		tmp = (l / F) * (math.pi / -F)
	elif l <= 0.5:
		tmp = t_0
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))))
	tmp = 0.0
	if (l <= -1.5)
		tmp = Float64(pi * l);
	elseif (l <= -1.1e-293)
		tmp = t_0;
	elseif (l <= 5.6e-280)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif (l <= 0.5)
		tmp = t_0;
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = l * (pi * (1.0 - (F ^ -2.0)));
	tmp = 0.0;
	if (l <= -1.5)
		tmp = pi * l;
	elseif (l <= -1.1e-293)
		tmp = t_0;
	elseif (l <= 5.6e-280)
		tmp = (l / F) * (pi / -F);
	elseif (l <= 0.5)
		tmp = t_0;
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.5], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -1.1e-293], t$95$0, If[LessEqual[l, 5.6e-280], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.5], t$95$0, N[(Pi * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.5 or 0.5 < l

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -1.5 < l < -1.1e-293 or 5.60000000000000035e-280 < l < 0.5

   1. Initial program 90.8%

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

      1. associate-*l/ 90.8%

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

      2. *-lft-identity 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in l around 0 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in l around 0 90.7%

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

      1. unpow2 90.7%

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

      2. *-lft-identity 90.7%

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

      3. times-frac 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-*l/ 90.6%

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

      6. associate-*l* 90.6%

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

      7. unpow-1 90.6%

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

      8. unpow-1 90.6%

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

      9. pow-sqr 90.7%

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

      10. metadata-eval 90.7%

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

      11. *-commutative 90.7%

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

      12. *-rgt-identity 90.7%

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

      13. distribute-lft-out-- 90.6%

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

   9. Simplified 90.6%

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

   if -1.1e-293 < l < 5.60000000000000035e-280

   1. Initial program 46.0%

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

      1. associate-*l/ 46.0%

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

      2. *-lft-identity 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in l around 0 46.1%

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

      1. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in F around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. associate-/l* 26.9%

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

      3. unpow2 26.9%

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

      4. associate-*r/ 27.1%

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

      5. distribute-neg-frac 27.1%

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

      6. associate-*r/ 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in l around 0 26.9%

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

       1. unpow2 26.9%

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

       2. associate-*r/ 26.9%

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

       3. *-commutative 26.9%

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

       4. associate-*l/ 26.9%

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

       5. associate-*r* 26.9%

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

       6. associate-/r* 27.0%

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

       7. metadata-eval 27.0%

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

       8. associate-/r* 27.0%

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

       9. neg-mul-1 27.0%

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

       10. associate-/r* 26.9%

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

       11. *-commutative 26.9%

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

       12. associate-/r/ 26.8%

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

       13. associate-*l/ 26.9%

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

       14. *-lft-identity 26.9%

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

       15. associate-/l* 27.0%

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

       16. associate-/r/ 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-293}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 10: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-293}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -1.5)
   (* PI l)
   (if (<= l -1.05e-293)
     (* l (* PI (- 1.0 (pow F -2.0))))
     (if (<= l 5.6e-280)
       (* (/ l F) (/ PI (- F)))
       (if (<= l 0.5) (* l (- PI (/ PI (* F F)))) (* PI l))))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -1.5) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -1.05e-293) {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	} else if (l <= 5.6e-280) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if (l <= 0.5) {
		tmp = l * (((double) M_PI) - (((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 (l <= -1.5) {
		tmp = Math.PI * l;
	} else if (l <= -1.05e-293) {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	} else if (l <= 5.6e-280) {
		tmp = (l / F) * (Math.PI / -F);
	} else if (l <= 0.5) {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -1.5:
		tmp = math.pi * l
	elif l <= -1.05e-293:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	elif l <= 5.6e-280:
		tmp = (l / F) * (math.pi / -F)
	elif l <= 0.5:
		tmp = l * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -1.5)
		tmp = Float64(pi * l);
	elseif (l <= -1.05e-293)
		tmp = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	elseif (l <= 5.6e-280)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif (l <= 0.5)
		tmp = Float64(l * Float64(pi - 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 (l <= -1.5)
		tmp = pi * l;
	elseif (l <= -1.05e-293)
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	elseif (l <= 5.6e-280)
		tmp = (l / F) * (pi / -F);
	elseif (l <= 0.5)
		tmp = l * (pi - (pi / (F * F)));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -1.5], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -1.05e-293], N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e-280], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.5], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.5 or 0.5 < l

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -1.5 < l < -1.05000000000000003e-293

   1. Initial program 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

      2. *-lft-identity 88.6%

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

      3. times-frac 88.6%

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

      4. *-lft-identity 88.6%

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

      5. associate-*l/ 88.6%

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

      6. associate-*l* 88.5%

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

      7. unpow-1 88.5%

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

      8. unpow-1 88.5%

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

      9. pow-sqr 88.6%

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

      10. metadata-eval 88.6%

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

      11. *-commutative 88.6%

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

      12. *-rgt-identity 88.6%

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

      13. distribute-lft-out-- 88.6%

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

   9. Simplified 88.6%

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

   if -1.05000000000000003e-293 < l < 5.60000000000000035e-280

   1. Initial program 46.0%

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

      1. associate-*l/ 46.0%

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

      2. *-lft-identity 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in l around 0 46.1%

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

      1. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in F around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. associate-/l* 26.9%

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

      3. unpow2 26.9%

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

      4. associate-*r/ 27.1%

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

      5. distribute-neg-frac 27.1%

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

      6. associate-*r/ 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in l around 0 26.9%

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

       1. unpow2 26.9%

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

       2. associate-*r/ 26.9%

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

       3. *-commutative 26.9%

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

       4. associate-*l/ 26.9%

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

       5. associate-*r* 26.9%

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

       6. associate-/r* 27.0%

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

       7. metadata-eval 27.0%

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

       8. associate-/r* 27.0%

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

       9. neg-mul-1 27.0%

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

       10. associate-/r* 26.9%

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

       11. *-commutative 26.9%

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

       12. associate-/r/ 26.8%

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

       13. associate-*l/ 26.9%

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

       14. *-lft-identity 26.9%

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

       15. associate-/l* 27.0%

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

       16. associate-/r/ 80.4%

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

   12. Simplified 80.4%

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

   if 5.60000000000000035e-280 < l < 0.5

   1. Initial program 93.1%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in l around 0 92.8%

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

      1. unpow2 92.8%

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

   6. Simplified 92.8%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-293}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 11: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {F}^{-2}\\ \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-293}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \left(\ell \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow F -2.0))))
   (if (<= l -1.5)
     (* PI l)
     (if (<= l -1.05e-293)
       (* l (* PI t_0))
       (if (<= l 8.4e-280)
         (* (/ l F) (/ PI (- F)))
         (if (<= l 0.5) (* PI (* l t_0)) (* PI l)))))))

C

double code(double F, double l) {
	double t_0 = 1.0 - pow(F, -2.0);
	double tmp;
	if (l <= -1.5) {
		tmp = ((double) M_PI) * l;
	} else if (l <= -1.05e-293) {
		tmp = l * (((double) M_PI) * t_0);
	} else if (l <= 8.4e-280) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if (l <= 0.5) {
		tmp = ((double) M_PI) * (l * t_0);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = 1.0 - Math.pow(F, -2.0);
	double tmp;
	if (l <= -1.5) {
		tmp = Math.PI * l;
	} else if (l <= -1.05e-293) {
		tmp = l * (Math.PI * t_0);
	} else if (l <= 8.4e-280) {
		tmp = (l / F) * (Math.PI / -F);
	} else if (l <= 0.5) {
		tmp = Math.PI * (l * t_0);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = 1.0 - math.pow(F, -2.0)
	tmp = 0
	if l <= -1.5:
		tmp = math.pi * l
	elif l <= -1.05e-293:
		tmp = l * (math.pi * t_0)
	elif l <= 8.4e-280:
		tmp = (l / F) * (math.pi / -F)
	elif l <= 0.5:
		tmp = math.pi * (l * t_0)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(1.0 - (F ^ -2.0))
	tmp = 0.0
	if (l <= -1.5)
		tmp = Float64(pi * l);
	elseif (l <= -1.05e-293)
		tmp = Float64(l * Float64(pi * t_0));
	elseif (l <= 8.4e-280)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif (l <= 0.5)
		tmp = Float64(pi * Float64(l * t_0));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = 1.0 - (F ^ -2.0);
	tmp = 0.0;
	if (l <= -1.5)
		tmp = pi * l;
	elseif (l <= -1.05e-293)
		tmp = l * (pi * t_0);
	elseif (l <= 8.4e-280)
		tmp = (l / F) * (pi / -F);
	elseif (l <= 0.5)
		tmp = pi * (l * t_0);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.5], N[(Pi * l), $MachinePrecision], If[LessEqual[l, -1.05e-293], N[(l * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.4e-280], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.5], N[(Pi * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.5 or 0.5 < l

   1. Initial program 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -1.5 < l < -1.05000000000000003e-293

   1. Initial program 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in l around 0 88.6%

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

      1. unpow2 88.6%

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

      2. *-lft-identity 88.6%

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

      3. times-frac 88.6%

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

      4. *-lft-identity 88.6%

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

      5. associate-*l/ 88.6%

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

      6. associate-*l* 88.5%

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

      7. unpow-1 88.5%

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

      8. unpow-1 88.5%

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

      9. pow-sqr 88.6%

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

      10. metadata-eval 88.6%

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

      11. *-commutative 88.6%

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

      12. *-rgt-identity 88.6%

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

      13. distribute-lft-out-- 88.6%

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

   9. Simplified 88.6%

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

   if -1.05000000000000003e-293 < l < 8.40000000000000003e-280

   1. Initial program 46.0%

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

      1. associate-*l/ 46.0%

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

      2. *-lft-identity 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in l around 0 46.1%

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

      1. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in F around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. associate-/l* 26.9%

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

      3. unpow2 26.9%

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

      4. associate-*r/ 27.1%

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

      5. distribute-neg-frac 27.1%

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

      6. associate-*r/ 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in l around 0 26.9%

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

       1. unpow2 26.9%

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

       2. associate-*r/ 26.9%

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

       3. *-commutative 26.9%

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

       4. associate-*l/ 26.9%

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

       5. associate-*r* 26.9%

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

       6. associate-/r* 27.0%

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

       7. metadata-eval 27.0%

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

       8. associate-/r* 27.0%

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

       9. neg-mul-1 27.0%

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

       10. associate-/r* 26.9%

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

       11. *-commutative 26.9%

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

       12. associate-/r/ 26.8%

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

       13. associate-*l/ 26.9%

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

       14. *-lft-identity 26.9%

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

       15. associate-/l* 27.0%

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

       16. associate-/r/ 80.4%

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

   12. Simplified 80.4%

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

   if 8.40000000000000003e-280 < l < 0.5

   1. Initial program 93.1%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in l around 0 92.8%

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

      1. unpow2 92.8%

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

   6. Simplified 92.8%

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

      1. add-sqr-sqrt 52.1%

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

      2. pow2 52.1%

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

      3. sub-neg 52.1%

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

      4. sub-neg 52.1%

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

      5. div-inv 52.1%

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

      6. pow2 52.1%

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

      7. pow-flip 52.1%

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

      8. metadata-eval 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. unpow2 52.1%

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

      2. add-sqr-sqrt 92.8%

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

      3. *-commutative 92.8%

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

      4. *-un-lft-identity 92.8%

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

      5. distribute-rgt-out-- 92.7%

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

      6. *-commutative 92.7%

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

      7. associate-*r* 92.8%

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

   10. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(1 - {F}^{-2}\right)\right) \cdot \pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-293}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \left(\ell \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 12: 75.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-292} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-236}\right) \land F \cdot F \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* F F) 5e-292)
         (and (not (<= (* F F) 2e-236)) (<= (* F F) 2e-49)))
   (* (/ l F) (/ PI (- F)))
   (* PI l)))

C

double code(double F, double l) {
	double tmp;
	if (((F * F) <= 5e-292) || (!((F * F) <= 2e-236) && ((F * F) <= 2e-49))) {
		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) <= 5e-292) || (!((F * F) <= 2e-236) && ((F * F) <= 2e-49))) {
		tmp = (l / F) * (Math.PI / -F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((F * F) <= 5e-292) or (not ((F * F) <= 2e-236) and ((F * F) <= 2e-49)):
		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) <= 5e-292) || (!(Float64(F * F) <= 2e-236) && (Float64(F * F) <= 2e-49)))
		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) <= 5e-292) || (~(((F * F) <= 2e-236)) && ((F * F) <= 2e-49)))
		tmp = (l / F) * (pi / -F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(F * F), $MachinePrecision], 5e-292], And[N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-236]], $MachinePrecision], LessEqual[N[(F * F), $MachinePrecision], 2e-49]]], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 4.99999999999999981e-292 or 2.0000000000000001e-236 < (*.f64 F F) < 1.99999999999999987e-49

   1. Initial program 53.7%

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

      1. associate-*l/ 53.7%

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

      2. *-lft-identity 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in l around 0 45.1%

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

      1. unpow2 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in F around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. associate-/l* 45.2%

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

      3. unpow2 45.2%

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

      4. associate-*r/ 45.1%

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

      5. distribute-neg-frac 45.1%

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

      6. associate-*r/ 45.2%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in l around 0 45.1%

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

       1. unpow2 45.1%

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

       2. associate-*r/ 45.1%

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

       3. *-commutative 45.1%

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

       4. associate-*l/ 45.1%

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

       5. associate-*r* 45.1%

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

       6. associate-/r* 45.1%

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

       7. metadata-eval 45.1%

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

       8. associate-/r* 45.1%

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

       9. neg-mul-1 45.1%

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

       10. associate-/r* 45.1%

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

       11. *-commutative 45.1%

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

       12. associate-/r/ 45.1%

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

       13. associate-*l/ 45.2%

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

       14. *-lft-identity 45.2%

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

       15. associate-/l* 45.1%

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

       16. associate-/r/ 63.4%

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

   12. Simplified 63.4%

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

   if 4.99999999999999981e-292 < (*.f64 F F) < 2.0000000000000001e-236 or 1.99999999999999987e-49 < (*.f64 F F)

   1. Initial program 94.8%

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

      1. associate-*l/ 94.8%

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

      2. *-lft-identity 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in l around 0 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in F around inf 95.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-292} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-236}\right) \land F \cdot F \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 13: 75.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-236} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 5e-292)
   (* (/ l F) (/ PI (- F)))
   (if (or (<= (* F F) 2e-236) (not (<= (* F F) 2e-49)))
     (* PI l)
     (/ (- l) (/ (* F F) PI)))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 5e-292) {
		tmp = (l / F) * (((double) M_PI) / -F);
	} else if (((F * F) <= 2e-236) || !((F * F) <= 2e-49)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = -l / ((F * F) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 5e-292) {
		tmp = (l / F) * (Math.PI / -F);
	} else if (((F * F) <= 2e-236) || !((F * F) <= 2e-49)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / ((F * F) / Math.PI);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 5e-292:
		tmp = (l / F) * (math.pi / -F)
	elif ((F * F) <= 2e-236) or not ((F * F) <= 2e-49):
		tmp = math.pi * l
	else:
		tmp = -l / ((F * F) / math.pi)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 5e-292)
		tmp = Float64(Float64(l / F) * Float64(pi / Float64(-F)));
	elseif ((Float64(F * F) <= 2e-236) || !(Float64(F * F) <= 2e-49))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 5e-292)
		tmp = (l / F) * (pi / -F);
	elseif (((F * F) <= 2e-236) || ~(((F * F) <= 2e-49)))
		tmp = pi * l;
	else
		tmp = -l / ((F * F) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 5e-292], N[(N[(l / F), $MachinePrecision] * N[(Pi / (-F)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(F * F), $MachinePrecision], 2e-236], N[Not[LessEqual[N[(F * F), $MachinePrecision], 2e-49]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 4.99999999999999981e-292

   1. Initial program 34.0%

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

      1. associate-*l/ 34.0%

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

      2. *-lft-identity 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in l around 0 29.3%

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

      1. unpow2 29.3%

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

   6. Simplified 29.3%

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

   7. Taylor expanded in F around 0 29.3%

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

      1. mul-1-neg 29.3%

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

      2. associate-/l* 29.3%

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

      3. unpow2 29.3%

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

      4. associate-*r/ 29.3%

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

      5. distribute-neg-frac 29.3%

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

      6. associate-*r/ 29.3%

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

   9. Simplified 29.3%

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

   10. Taylor expanded in l around 0 29.3%

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

       1. unpow2 29.3%

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

       2. associate-*r/ 29.3%

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

       3. *-commutative 29.3%

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

       4. associate-*l/ 29.3%

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

       5. associate-*r* 29.3%

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

       6. associate-/r* 29.3%

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

       7. metadata-eval 29.3%

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

       8. associate-/r* 29.3%

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

       9. neg-mul-1 29.3%

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

       10. associate-/r* 29.3%

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

       11. *-commutative 29.3%

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

       12. associate-/r/ 29.3%

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

       13. associate-*l/ 29.3%

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

       14. *-lft-identity 29.3%

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

       15. associate-/l* 29.3%

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

       16. associate-/r/ 57.9%

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

   12. Simplified 57.9%

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

   if 4.99999999999999981e-292 < (*.f64 F F) < 2.0000000000000001e-236 or 1.99999999999999987e-49 < (*.f64 F F)

   1. Initial program 94.8%

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

      1. associate-*l/ 94.8%

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

      2. *-lft-identity 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in l around 0 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in F around inf 95.3%

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

   if 2.0000000000000001e-236 < (*.f64 F F) < 1.99999999999999987e-49

   1. Initial program 88.5%

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

      1. associate-*l/ 88.6%

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

      2. *-lft-identity 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in l around 0 73.1%

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

      1. unpow2 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in F around 0 73.2%

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

      1. mul-1-neg 73.2%

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

      2. associate-/l* 73.3%

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

      3. unpow2 73.3%

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

      4. associate-*r/ 73.1%

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

      5. distribute-neg-frac 73.1%

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

      6. associate-*r/ 73.3%

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

   9. Simplified 73.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-236} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 14: 73.3% 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 79.2%

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

   1. associate-*l/ 79.2%

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

   2. *-lft-identity 79.2%

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

3. Simplified 79.2%

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

4. Taylor expanded in l around 0 73.4%

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

   1. unpow2 73.4%

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

6. Simplified 73.4%

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

7. Taylor expanded in F around inf 72.5%

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

8. Final simplification 72.5%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))