VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.3%

Time: 19.1s

Alternatives: 10

Speedup: 1.5×

• could not determine a ground truth

  1. F = -1.1768699212260474e+125
  2. l = 1601593503066044.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+17) (not (<= (* PI l) 1000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+17) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+17) || !((Math.PI * l) <= 1000000000000.0)) {
		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) <= -5e+17) or not ((math.pi * l) <= 1000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+17) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+17) || ~(((pi * l) <= 1000000000000.0)))
		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], -5e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e17 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 63.6%

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

      1. associate-*l/ 63.6%

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

      2. *-lft-identity 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -5e17 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 93.2%

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

      1. associate-*l/ 93.1%

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

      2. *-un-lft-identity 93.1%

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

      3. associate-/r* 98.5%

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

   3. Applied egg-rr 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200 \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -200.0) (not (<= (* PI l) 0.01)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -200.0) || !((((double) M_PI) * l) <= 0.01)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / (F / ((double) M_PI))) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -200.0) || !((Math.PI * l) <= 0.01)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / (F / Math.PI)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -200.0) or not ((math.pi * l) <= 0.01):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / (F / math.pi)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -200.0) || !(Float64(pi * l) <= 0.01))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / Float64(F / pi)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -200.0) || ~(((pi * l) <= 0.01)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / (F / pi)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -200.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -200 or 0.0100000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in l around 0 53.6%

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

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -200 < (*.f64 (PI.f64) l) < 0.0100000000000000002

   1. Initial program 93.8%

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

      1. associate-*l/ 93.8%

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

      2. *-un-lft-identity 93.8%

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

      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 99.5%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200 \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 18.5)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		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 ((l <= -2.5) || !(l <= 18.5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -2.5) or not (l <= 18.5):
		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 ((l <= -2.5) || !(l <= 18.5))
		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 ((l <= -2.5) || ~((l <= 18.5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -2.5], N[Not[LessEqual[l, 18.5]], $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 l < -2.5 or 18.5 < l

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in l around 0 53.6%

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

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -2.5 < l < 18.5

   1. Initial program 93.8%

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

      1. add-sqr-sqrt 93.7%

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

      2. sqrt-unprod 83.6%

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

      3. inv-pow 83.6%

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

      4. unpow-prod-down 83.5%

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

      5. pow-prod-up 83.6%

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

      6. inv-pow 83.6%

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

      7. unpow-prod-down 83.6%

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

      8. pow-prod-up 83.6%

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

      9. pow-prod-up 83.6%

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

      10. metadata-eval 83.6%

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

      11. metadata-eval 83.6%

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

      12. metadata-eval 83.6%

          \[\leadsto \pi \cdot \ell - \sqrt{{F}^{\color{blue}{-4}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Applied egg-rr 83.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt{{F}^{-4}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   4. Taylor expanded in l around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. unpow2 93.8%

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

      3. times-frac 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 18.5)))
   (* PI l)
   (- (* PI l) (/ (* l (/ PI F)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l * (((double) M_PI) / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l * (Math.PI / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -2.5) or not (l <= 18.5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l * (math.pi / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -2.5) || !(l <= 18.5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l * Float64(pi / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -2.5) || ~((l <= 18.5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l * (pi / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -2.5], N[Not[LessEqual[l, 18.5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5 or 18.5 < l

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in l around 0 53.6%

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

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -2.5 < l < 18.5

   1. Initial program 93.8%

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

      1. associate-*l/ 93.8%

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

      2. *-un-lft-identity 93.8%

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

      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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. associate-*r/ 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \]

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 18.5)))
   (* PI l)
   (* l (* PI (- 1.0 (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		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 <= -2.5) || !(l <= 18.5)) {
		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 <= -2.5) or not (l <= 18.5):
		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 <= -2.5) || !(l <= 18.5))
		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 <= -2.5) || ~((l <= 18.5)))
		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, -2.5], N[Not[LessEqual[l, 18.5]], $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 < -2.5 or 18.5 < l

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in l around 0 53.6%

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

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -2.5 < l < 18.5

   1. Initial program 93.8%

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

      1. associate-*l/ 93.8%

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

      2. *-lft-identity 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in l around 0 93.8%

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

      1. unpow2 93.8%

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

   6. Simplified 93.8%

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

      1. sub-neg 93.8%

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

      2. distribute-lft-in 93.8%

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

      3. *-commutative 93.8%

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

      4. clear-num 93.8%

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

      5. associate-/r/ 93.8%

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

      6. pow2 93.8%

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

      7. pow-flip 93.8%

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

      8. metadata-eval 93.8%

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

      9. metadata-eval 93.8%

         \[\leadsto \pi \cdot \ell + \ell \cdot \left(-{F}^{\color{blue}{\left(\frac{-4}{2}\right)}} \cdot \pi\right) \]

      10. sqrt-pow1 83.6%

          \[\leadsto \pi \cdot \ell + \ell \cdot \left(-\color{blue}{\sqrt{{F}^{-4}}} \cdot \pi\right) \]

      11. *-commutative 83.6%

          \[\leadsto \pi \cdot \ell + \ell \cdot \left(-\color{blue}{\pi \cdot \sqrt{{F}^{-4}}}\right) \]

      12. distribute-lft-neg-in 83.6%

          \[\leadsto \pi \cdot \ell + \ell \cdot \color{blue}{\left(\left(-\pi\right) \cdot \sqrt{{F}^{-4}}\right)} \]

      13. sqrt-pow1 93.8%

          \[\leadsto \pi \cdot \ell + \ell \cdot \left(\left(-\pi\right) \cdot \color{blue}{{F}^{\left(\frac{-4}{2}\right)}}\right) \]

      14. metadata-eval 93.8%

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

   8. Applied egg-rr 93.8%

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

      1. *-commutative 93.8%

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

      2. distribute-rgt-out 93.8%

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

      3. cancel-sign-sub-inv 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. distribute-lft-out-- 93.8%

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

   10. Simplified 93.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 18.5))) (* PI l) (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -2.5) || !(l <= 18.5)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -2.5) or not (l <= 18.5):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -2.5) || !(l <= 18.5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -2.5) || ~((l <= 18.5)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -2.5], N[Not[LessEqual[l, 18.5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5 or 18.5 < l

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in l around 0 53.6%

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

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in F around inf 97.3%

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

   if -2.5 < l < 18.5

   1. Initial program 93.8%

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

      1. associate-*l/ 93.8%

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

      2. *-lft-identity 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in l around 0 93.8%

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

      1. unpow2 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 18.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 73.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 4.4 \cdot 10^{-26} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 4.4e-26) (not (<= F 0.88))) (* PI l) (/ (- l) (* F (/ F PI)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 4.4e-26) || !(F <= 0.88)) {
		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 <= 4.4e-26) || !(F <= 0.88)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / (F * (F / Math.PI));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 4.4e-26) or not (F <= 0.88):
		tmp = math.pi * l
	else:
		tmp = -l / (F * (F / math.pi))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 4.4e-26) || !(F <= 0.88))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(F * Float64(F / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 4.4e-26) || ~((F <= 0.88)))
		tmp = pi * l;
	else
		tmp = -l / (F * (F / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 4.4e-26], N[Not[LessEqual[F, 0.88]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(F * N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.4000000000000002e-26 or 0.880000000000000004 < F

   1. Initial program 79.3%

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

      1. associate-*l/ 79.3%

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

      2. *-lft-identity 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in l around 0 73.6%

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

      1. unpow2 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 4.4000000000000002e-26 < F < 0.880000000000000004

   1. Initial program 99.0%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in l around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

      1. add-cbrt-cube 42.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}} \]

      2. pow3 42.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}^{3}}} \]

      3. sub-neg 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \color{blue}{\left(\pi + \left(-\frac{\pi}{F \cdot F}\right)\right)}\right)}^{3}} \]

      4. sub-neg 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \color{blue}{\left(\pi - \frac{\pi}{F \cdot F}\right)}\right)}^{3}} \]

      5. div-inv 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \color{blue}{\pi \cdot \frac{1}{F \cdot F}}\right)\right)}^{3}} \]

      6. pow2 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right)}^{3}} \]

      7. pow-flip 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right)}^{3}} \]

      8. metadata-eval 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot {F}^{\color{blue}{-2}}\right)\right)}^{3}} \]

   8. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot {F}^{-2}\right)\right)}^{3}}} \]

   9. Taylor expanded in F around 0 88.9%

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

       1. mul-1-neg 88.9%

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

       2. associate-/l* 88.4%

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

       3. unpow2 88.4%

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

   11. Simplified 88.4%

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

   12. Taylor expanded in F around 0 88.4%

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

       1. unpow2 88.4%

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

       2. associate-*r/ 88.4%

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

   14. Simplified 88.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.4 \cdot 10^{-26} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \end{array} \]

Alternative 8: 73.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.18 \cdot 10^{-26} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 1.18e-26) (not (<= F 0.88))) (* PI l) (/ (- l) (/ (* F F) PI))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 1.18e-26) || !(F <= 0.88)) {
		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 <= 1.18e-26) || !(F <= 0.88)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / ((F * F) / Math.PI);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 1.18e-26) or not (F <= 0.88):
		tmp = math.pi * l
	else:
		tmp = -l / ((F * F) / math.pi)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 1.18e-26) || !(F <= 0.88))
		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 <= 1.18e-26) || ~((F <= 0.88)))
		tmp = pi * l;
	else
		tmp = -l / ((F * F) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 1.18e-26], N[Not[LessEqual[F, 0.88]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.17999999999999996e-26 or 0.880000000000000004 < F

   1. Initial program 79.3%

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

      1. associate-*l/ 79.3%

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

      2. *-lft-identity 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in l around 0 73.6%

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

      1. unpow2 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 1.17999999999999996e-26 < F < 0.880000000000000004

   1. Initial program 99.0%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in l around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

      1. add-cbrt-cube 42.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}} \]

      2. pow3 42.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}^{3}}} \]

      3. sub-neg 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \color{blue}{\left(\pi + \left(-\frac{\pi}{F \cdot F}\right)\right)}\right)}^{3}} \]

      4. sub-neg 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \color{blue}{\left(\pi - \frac{\pi}{F \cdot F}\right)}\right)}^{3}} \]

      5. div-inv 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \color{blue}{\pi \cdot \frac{1}{F \cdot F}}\right)\right)}^{3}} \]

      6. pow2 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right)}^{3}} \]

      7. pow-flip 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right)}^{3}} \]

      8. metadata-eval 42.5%

         \[\leadsto \sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot {F}^{\color{blue}{-2}}\right)\right)}^{3}} \]

   8. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\ell \cdot \left(\pi - \pi \cdot {F}^{-2}\right)\right)}^{3}}} \]

   9. Taylor expanded in F around 0 88.9%

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

       1. mul-1-neg 88.9%

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

       2. associate-/l* 88.4%

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

       3. unpow2 88.4%

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

   11. Simplified 88.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.18 \cdot 10^{-26} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 9: 73.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.22 \cdot 10^{-25} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 1.22e-25) (not (<= F 0.88))) (* PI l) (/ (* PI (- l)) (* F F))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 1.22e-25) || !(F <= 0.88)) {
		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 ((F <= 1.22e-25) || !(F <= 0.88)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * -l) / (F * F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 1.22e-25) or not (F <= 0.88):
		tmp = math.pi * l
	else:
		tmp = (math.pi * -l) / (F * F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 1.22e-25) || !(F <= 0.88))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * Float64(-l)) / Float64(F * F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 1.22e-25) || ~((F <= 0.88)))
		tmp = pi * l;
	else
		tmp = (pi * -l) / (F * F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 1.22e-25], N[Not[LessEqual[F, 0.88]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * (-l)), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.21999999999999999e-25 or 0.880000000000000004 < F

   1. Initial program 79.3%

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

      1. associate-*l/ 79.3%

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

      2. *-lft-identity 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in l around 0 73.6%

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

      1. unpow2 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 1.21999999999999999e-25 < F < 0.880000000000000004

   1. Initial program 99.0%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in l around 0 98.7%

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

      1. unpow2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in F around 0 88.9%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. mul-1-neg 88.9%

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

      4. *-commutative 88.9%

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

      5. unpow2 88.9%

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

   9. Simplified 88.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.22 \cdot 10^{-25} \lor \neg \left(F \leq 0.88\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 10: 73.7% 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 79.8%

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

   1. associate-*l/ 79.8%

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

   2. *-lft-identity 79.8%

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

3. Simplified 79.8%

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

4. Taylor expanded in l around 0 74.2%

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

   1. unpow2 74.2%

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

6. Simplified 74.2%

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

7. Taylor expanded in F around inf 77.1%

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

8. Final simplification 77.1%

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

Reproduce

herbie shell --seed 2023256 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))