VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.1%

Time: 19.7s

Alternatives: 7

Speedup: 1.5×

• could not determine a ground truth

  1. F = 6.500473520491322e-229
  2. l = -2599415327828500.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 40000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+16) (not (<= (* PI l) 40000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+16) || !((((double) M_PI) * l) <= 40000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+16) || !((Math.PI * l) <= 40000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+16) or not ((math.pi * l) <= 40000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+16) || !(Float64(pi * l) <= 40000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+16) || ~(((pi * l) <= 40000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 40000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e16 or 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 64.5%

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

      1. associate-*l/ 64.5%

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

      2. *-lft-identity 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in l around 0 50.1%

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

      1. unpow2 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e16 < (*.f64 (PI.f64) l) < 4e7

   1. Initial program 90.2%

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

      1. associate-*l/ 90.4%

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

      2. *-un-lft-identity 90.4%

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

      3. associate-/r* 98.7%

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

   3. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 40000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000000000 \lor \neg \left(\pi \cdot \ell \leq 40000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -100000000000.0) (not (<= (* PI l) 40000000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -100000000000.0) || !((((double) M_PI) * l) <= 40000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -100000000000.0) || !((Math.PI * l) <= 40000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -100000000000.0) or not ((math.pi * l) <= 40000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -100000000000.0) || !(Float64(pi * l) <= 40000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -100000000000.0) || ~(((pi * l) <= 40000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -100000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 40000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e11 or 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 64.3%

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

      1. associate-*l/ 64.3%

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

      2. *-lft-identity 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in l around 0 49.7%

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

      1. unpow2 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -1e11 < (*.f64 (PI.f64) l) < 4e7

   1. Initial program 90.6%

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

      1. inv-pow 90.6%

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

      2. unpow-prod-down 90.5%

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

      3. inv-pow 90.5%

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

      4. inv-pow 90.5%

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

   3. Applied egg-rr 90.5%

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

   4. Taylor expanded in l around 0 90.0%

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

      1. *-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. times-frac 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000000000 \lor \neg \left(\pi \cdot \ell \leq 40000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1450000000 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1450000000.0) (not (<= l 1e+15)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1450000000.0) || !(l <= 1e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1450000000.0) || !(l <= 1e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1450000000.0) or not (l <= 1e+15):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1450000000.0) || !(l <= 1e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1450000000.0) || ~((l <= 1e+15)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1450000000.0], N[Not[LessEqual[l, 1e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.45e9 or 1e15 < l

   1. Initial program 64.3%

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

      1. associate-*l/ 64.3%

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

      2. *-lft-identity 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in l around 0 49.7%

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

      1. unpow2 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -1.45e9 < l < 1e15

   1. Initial program 90.6%

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

   2. Taylor expanded in l around 0 90.0%

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

      1. associate-/l* 90.0%

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

      2. associate-/r/ 90.0%

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

      3. unpow2 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1450000000 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4: 92.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1450000000 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1450000000.0) (not (<= l 1e+15)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1450000000.0) || !(l <= 1e+15)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1450000000.0) || !(l <= 1e+15)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1450000000.0) or not (l <= 1e+15):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi * (1.0 - math.pow(F, -2.0)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1450000000.0) || !(l <= 1e+15))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1450000000.0) || ~((l <= 1e+15)))
		tmp = pi * l;
	else
		tmp = l * (pi * (1.0 - (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1450000000.0], N[Not[LessEqual[l, 1e+15]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.45e9 or 1e15 < l

   1. Initial program 64.3%

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

      1. associate-*l/ 64.3%

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

      2. *-lft-identity 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in l around 0 49.7%

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

      1. unpow2 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in F around inf 98.8%

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

   if -1.45e9 < l < 1e15

   1. Initial program 90.6%

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

      1. associate-*l/ 90.8%

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

      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 88.4%

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

      1. unpow2 88.4%

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

   6. Simplified 88.4%

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

      1. sub-neg 88.4%

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

      2. distribute-lft-in 88.4%

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

      3. *-commutative 88.4%

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

      4. div-inv 88.4%

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

      5. metadata-eval 88.4%

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

      6. frac-times 88.4%

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

      7. distribute-rgt-neg-in 88.4%

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

      8. inv-pow 88.4%

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

      9. inv-pow 88.4%

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

      10. pow-prod-up 88.4%

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

      11. metadata-eval 88.4%

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

   8. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. distribute-rgt-out 88.4%

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

      3. distribute-rgt-neg-out 88.4%

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

      4. sub-neg 88.4%

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

      5. *-rgt-identity 88.4%

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

      6. distribute-lft-out-- 88.4%

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

   10. Simplified 88.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1450000000 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 5: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.008 \lor \neg \left(\ell \leq -3 \cdot 10^{-106} \lor \neg \left(\ell \leq 3.5 \cdot 10^{-138}\right) \land \ell \leq 1.2 \cdot 10^{-27}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -0.008)
         (not (or (<= l -3e-106) (and (not (<= l 3.5e-138)) (<= l 1.2e-27)))))
   (* PI l)
   (* PI (/ (- l) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.008) || !((l <= -3e-106) || (!(l <= 3.5e-138) && (l <= 1.2e-27)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (-l / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -0.008) || !((l <= -3e-106) || (!(l <= 3.5e-138) && (l <= 1.2e-27)))) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (-l / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -0.008) or not ((l <= -3e-106) or (not (l <= 3.5e-138) and (l <= 1.2e-27))):
		tmp = math.pi * l
	else:
		tmp = math.pi * (-l / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -0.008) || !((l <= -3e-106) || (!(l <= 3.5e-138) && (l <= 1.2e-27))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(Float64(-l) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.008) || ~(((l <= -3e-106) || (~((l <= 3.5e-138)) && (l <= 1.2e-27)))))
		tmp = pi * l;
	else
		tmp = pi * (-l / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -0.008], N[Not[Or[LessEqual[l, -3e-106], And[N[Not[LessEqual[l, 3.5e-138]], $MachinePrecision], LessEqual[l, 1.2e-27]]]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[((-l) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0080000000000000002 or -3.00000000000000019e-106 < l < 3.4999999999999999e-138 or 1.20000000000000001e-27 < l

   1. Initial program 74.7%

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

      1. associate-*l/ 74.8%

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

      2. *-lft-identity 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in l around 0 65.4%

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

      1. unpow2 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in F around inf 80.1%

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

   if -0.0080000000000000002 < l < -3.00000000000000019e-106 or 3.4999999999999999e-138 < l < 1.20000000000000001e-27

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. unpow2 71.4%

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

   9. Simplified 71.4%

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

       1. associate-/l* 71.5%

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

       2. associate-/r/ 71.4%

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

   11. Applied egg-rr 71.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.008 \lor \neg \left(\ell \leq -3 \cdot 10^{-106} \lor \neg \left(\ell \leq 3.5 \cdot 10^{-138}\right) \land \ell \leq 1.2 \cdot 10^{-27}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \end{array} \]

Alternative 6: 75.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-27} \lor \neg \left(F \leq -1.15 \cdot 10^{-162}\right) \land \left(F \leq -1.75 \cdot 10^{-205} \lor \neg \left(F \leq 2.5 \cdot 10^{-111}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F -1.4e-27)
         (and (not (<= F -1.15e-162))
              (or (<= F -1.75e-205) (not (<= F 2.5e-111)))))
   (* PI l)
   (* (/ PI F) (/ (- l) F))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= -1.4e-27) || (!(F <= -1.15e-162) && ((F <= -1.75e-205) || !(F <= 2.5e-111)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) / F) * (-l / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= -1.4e-27) || (!(F <= -1.15e-162) && ((F <= -1.75e-205) || !(F <= 2.5e-111)))) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI / F) * (-l / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= -1.4e-27) or (not (F <= -1.15e-162) and ((F <= -1.75e-205) or not (F <= 2.5e-111))):
		tmp = math.pi * l
	else:
		tmp = (math.pi / F) * (-l / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= -1.4e-27) || (!(F <= -1.15e-162) && ((F <= -1.75e-205) || !(F <= 2.5e-111))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi / F) * Float64(Float64(-l) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= -1.4e-27) || (~((F <= -1.15e-162)) && ((F <= -1.75e-205) || ~((F <= 2.5e-111)))))
		tmp = pi * l;
	else
		tmp = (pi / F) * (-l / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, -1.4e-27], And[N[Not[LessEqual[F, -1.15e-162]], $MachinePrecision], Or[LessEqual[F, -1.75e-205], N[Not[LessEqual[F, 2.5e-111]], $MachinePrecision]]]], N[(Pi * l), $MachinePrecision], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.4e-27 or -1.1499999999999999e-162 < F < -1.75e-205 or 2.5000000000000001e-111 < F

   1. Initial program 91.9%

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

      1. associate-*l/ 91.9%

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

      2. *-lft-identity 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in l around 0 83.0%

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

      1. unpow2 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in F around inf 93.7%

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

   if -1.4e-27 < F < -1.1499999999999999e-162 or -1.75e-205 < F < 2.5000000000000001e-111

   1. Initial program 55.5%

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

      1. associate-*l/ 55.8%

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

      2. *-lft-identity 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in l around 0 49.6%

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

      1. unpow2 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in F around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. unpow2 52.0%

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

   9. Simplified 52.0%

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

       1. *-commutative 52.0%

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

       2. times-frac 63.7%

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

   11. Applied egg-rr 63.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-27} \lor \neg \left(F \leq -1.15 \cdot 10^{-162}\right) \land \left(F \leq -1.75 \cdot 10^{-205} \lor \neg \left(F \leq 2.5 \cdot 10^{-111}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \]

Alternative 7: 73.2% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (F l) :precision binary64 (* PI l))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[F_, l_] := N[(Pi * l), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-*l/ 78.8%

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

   2. *-lft-identity 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in l around 0 70.9%

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

   1. unpow2 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in F around inf 72.1%

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

8. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2023175 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))