VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 98.3%

Time: 25.9s

Alternatives: 11

Speedup: 1.5×

• could not determine a ground truth

  1. F = 5.588578734233136e-16
  2. l = -3339045700574735.5

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: 76.9% 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: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+37} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+37) (not (<= (* PI l) 2e+14)))
   (* PI l)
   (+ (* PI l) (/ -1.0 (/ F (/ (tan (* PI l)) F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+37) || !((((double) M_PI) * l) <= 2e+14)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + (-1.0 / (F / (tan((((double) M_PI) * l)) / F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+37) || !((Math.PI * l) <= 2e+14)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + (-1.0 / (F / (Math.tan((Math.PI * l)) / F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+37) or not ((math.pi * l) <= 2e+14):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + (-1.0 / (F / (math.tan((math.pi * l)) / F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+37) || !(Float64(pi * l) <= 2e+14))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(F / Float64(tan(Float64(pi * l)) / F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+37) || ~(((pi * l) <= 2e+14)))
		tmp = pi * l;
	else
		tmp = (pi * l) + (-1.0 / (F / (tan((pi * l)) / F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+37], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e+14]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1.99999999999999991e37 or 2e14 < (*.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 51.6%

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

      1. unpow2 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -1.99999999999999991e37 < (*.f64 (PI.f64) l) < 2e14

   1. Initial program 89.1%

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

      1. associate-/r/ 89.3%

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

      2. associate-/l* 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+37} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+37} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+37) (not (<= (* PI l) 2e+14)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+37) || !((((double) M_PI) * l) <= 2e+14)) {
		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+37) || !((Math.PI * l) <= 2e+14)) {
		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+37) or not ((math.pi * l) <= 2e+14):
		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+37) || !(Float64(pi * l) <= 2e+14))
		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+37) || ~(((pi * l) <= 2e+14)))
		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+37], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e+14]], $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) < -1.99999999999999991e37 or 2e14 < (*.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 51.6%

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

      1. unpow2 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -1.99999999999999991e37 < (*.f64 (PI.f64) l) < 2e14

   1. Initial program 89.1%

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

      1. associate-*l/ 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. associate-/r* 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+37} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000000000000 \lor \neg \left(\pi \cdot \ell \leq 0.05\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -40000000000000.0) (not (<= (* PI l) 0.05)))
   (* PI l)
   (- (* PI l) (/ (/ (* PI l) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -40000000000000.0) || !((((double) M_PI) * l) <= 0.05)) {
		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) <= -40000000000000.0) || !((Math.PI * l) <= 0.05)) {
		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) <= -40000000000000.0) or not ((math.pi * l) <= 0.05):
		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) <= -40000000000000.0) || !(Float64(pi * l) <= 0.05))
		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) <= -40000000000000.0) || ~(((pi * l) <= 0.05)))
		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], -40000000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.05]], $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) < -4e13 or 0.050000000000000003 < (*.f64 (PI.f64) l)

   1. Initial program 66.2%

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

      1. associate-*l/ 66.2%

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

      2. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.2%

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

   if -4e13 < (*.f64 (PI.f64) l) < 0.050000000000000003

   1. Initial program 89.1%

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

      1. associate-/r* 89.1%

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

      2. metadata-eval 89.1%

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

      3. add-sqr-sqrt 47.7%

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

      4. sqrt-prod 68.6%

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

      5. sqrt-div 68.6%

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

      6. associate-*l/ 68.6%

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

      7. sqrt-div 68.8%

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

      8. metadata-eval 68.8%

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

      9. sqrt-prod 54.9%

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

      10. add-sqr-sqrt 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in l around 0 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000000000000 \lor \neg \left(\pi \cdot \ell \leq 0.05\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 4: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -620000000000.0) (not (<= l 0.5)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -620000000000.0) || !(l <= 0.5)) {
		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 <= -620000000000.0) || !(l <= 0.5)) {
		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 <= -620000000000.0) or not (l <= 0.5):
		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 <= -620000000000.0) || !(l <= 0.5))
		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 <= -620000000000.0) || ~((l <= 0.5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -620000000000.0], N[Not[LessEqual[l, 0.5]], $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 < -6.2e11 or 0.5 < l

   1. Initial program 66.2%

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

      1. associate-*l/ 66.2%

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

      2. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.2%

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

   if -6.2e11 < l < 0.5

   1. Initial program 89.1%

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

   2. Taylor expanded in l around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 88.0%

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

      3. unpow2 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -620000000000.0) (not (<= l 0.5)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -620000000000.0) || !(l <= 0.5)) {
		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 <= -620000000000.0) || !(l <= 0.5)) {
		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 <= -620000000000.0) or not (l <= 0.5):
		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 <= -620000000000.0) || !(l <= 0.5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * Float64(l / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -620000000000.0) || ~((l <= 0.5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi * (l / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -620000000000.0], N[Not[LessEqual[l, 0.5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.2e11 or 0.5 < l

   1. Initial program 66.2%

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

      1. associate-*l/ 66.2%

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

      2. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.2%

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

   if -6.2e11 < l < 0.5

   1. Initial program 89.1%

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

      1. associate-/r* 89.1%

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

      2. metadata-eval 89.1%

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

      3. add-sqr-sqrt 47.7%

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

      4. sqrt-prod 68.6%

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

      5. sqrt-div 68.6%

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

      6. associate-*l/ 68.6%

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

      7. sqrt-div 68.8%

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

      8. metadata-eval 68.8%

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

      9. sqrt-prod 54.9%

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

      10. add-sqr-sqrt 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in l around 0 98.3%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 6: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -620000000000.0) (not (<= l 0.5)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -620000000000.0) || !(l <= 0.5)) {
		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 <= -620000000000.0) || !(l <= 0.5)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -620000000000.0) or not (l <= 0.5):
		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 <= -620000000000.0) || !(l <= 0.5))
		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 <= -620000000000.0) || ~((l <= 0.5)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -620000000000.0], N[Not[LessEqual[l, 0.5]], $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 < -6.2e11 or 0.5 < l

   1. Initial program 66.2%

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

      1. associate-*l/ 66.2%

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

      2. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.2%

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

   if -6.2e11 < l < 0.5

   1. Initial program 89.1%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \ell \cdot {F}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -620000000000.0) (not (<= l 0.5)))
   (* PI l)
   (* PI (- l (* l (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -620000000000.0) || !(l <= 0.5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (l - (l * pow(F, -2.0)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -620000000000.0) || !(l <= 0.5)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (l - (l * Math.pow(F, -2.0)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -620000000000.0) or not (l <= 0.5):
		tmp = math.pi * l
	else:
		tmp = math.pi * (l - (l * math.pow(F, -2.0)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -620000000000.0) || !(l <= 0.5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(l - Float64(l * (F ^ -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -620000000000.0) || ~((l <= 0.5)))
		tmp = pi * l;
	else
		tmp = pi * (l - (l * (F ^ -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -620000000000.0], N[Not[LessEqual[l, 0.5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(l - N[(l * N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.2e11 or 0.5 < l

   1. Initial program 66.2%

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

      1. associate-*l/ 66.2%

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

      2. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in l around 0 52.3%

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

      1. unpow2 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in F around inf 98.2%

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

   if -6.2e11 < l < 0.5

   1. Initial program 89.1%

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

      1. associate-*l/ 89.2%

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

      2. *-lft-identity 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in l around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

      1. associate-/r* 87.9%

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

      2. div-inv 87.8%

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

   8. Applied egg-rr 87.8%

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

   9. Taylor expanded in l around 0 87.8%

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

       1. unpow2 87.8%

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

       2. /-rgt-identity 87.8%

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

       3. associate-/l* 87.8%

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

       4. associate-/r/ 87.8%

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

       5. *-rgt-identity 87.8%

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

       6. associate-*l/ 87.8%

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

       7. associate-*r/ 87.8%

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

       8. associate-*r* 87.7%

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

       9. unpow-1 87.7%

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

       10. unpow-1 87.7%

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

       11. pow-sqr 87.9%

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

       12. metadata-eval 87.9%

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

       13. distribute-rgt-out-- 87.9%

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

       14. associate-*r* 87.8%

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

       15. distribute-lft-out-- 87.9%

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

       16. *-commutative 87.9%

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

   11. Simplified 87.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -620000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \ell \cdot {F}^{-2}\right)\\ \end{array} \]

Alternative 8: 74.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-280} \lor \neg \left(F \leq 8.2 \cdot 10^{-206}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 6.2e-280) (not (<= F 8.2e-206)))
   (* PI l)
   (* PI (* (/ l F) (/ -1.0 F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 6.2e-280) || !(F <= 8.2e-206)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * ((l / F) * (-1.0 / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 6.2e-280) || !(F <= 8.2e-206)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * ((l / F) * (-1.0 / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 6.2e-280) or not (F <= 8.2e-206):
		tmp = math.pi * l
	else:
		tmp = math.pi * ((l / F) * (-1.0 / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 6.2e-280) || !(F <= 8.2e-206))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(Float64(l / F) * Float64(-1.0 / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 6.2e-280) || ~((F <= 8.2e-206)))
		tmp = pi * l;
	else
		tmp = pi * ((l / F) * (-1.0 / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 6.2e-280], N[Not[LessEqual[F, 8.2e-206]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(N[(l / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.20000000000000042e-280 or 8.20000000000000032e-206 < F

   1. Initial program 78.4%

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

      1. associate-*l/ 78.5%

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

      2. *-lft-identity 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in l around 0 69.8%

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

      1. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 6.20000000000000042e-280 < F < 8.20000000000000032e-206

   1. Initial program 50.0%

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

      1. associate-*l/ 50.0%

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

      2. *-lft-identity 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in l around 0 49.4%

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

      1. unpow2 49.4%

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

   6. Simplified 49.4%

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

      1. associate-/r* 49.4%

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

      2. div-inv 49.4%

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

   8. Applied egg-rr 49.4%

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

   9. Taylor expanded in F around 0 49.4%

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

       1. mul-1-neg 49.4%

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

       2. unpow2 49.4%

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

       3. associate-/l* 49.4%

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

       4. associate-*r/ 49.4%

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

   11. Simplified 49.4%

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

       1. associate-/r* 76.2%

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

       2. associate-/l* 76.2%

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

       3. div-inv 76.3%

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

       4. *-commutative 76.3%

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

       5. associate-*l* 76.5%

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

   13. Applied egg-rr 76.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-280} \lor \neg \left(F \leq 8.2 \cdot 10^{-206}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \]

Alternative 9: 74.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-282} \lor \neg \left(F \leq 9 \cdot 10^{-208}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 2.8e-282) (not (<= F 9e-208)))
   (* PI l)
   (* (/ l F) (/ (- PI) F))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 2.8e-282) || !(F <= 9e-208)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l / F) * (-((double) M_PI) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 2.8e-282) || !(F <= 9e-208)) {
		tmp = Math.PI * l;
	} else {
		tmp = (l / F) * (-Math.PI / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 2.8e-282) or not (F <= 9e-208):
		tmp = math.pi * l
	else:
		tmp = (l / F) * (-math.pi / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 2.8e-282) || !(F <= 9e-208))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l / F) * Float64(Float64(-pi) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 2.8e-282) || ~((F <= 9e-208)))
		tmp = pi * l;
	else
		tmp = (l / F) * (-pi / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 2.8e-282], N[Not[LessEqual[F, 9e-208]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.7999999999999999e-282 or 8.9999999999999992e-208 < F

   1. Initial program 78.4%

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

      1. associate-*l/ 78.5%

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

      2. *-lft-identity 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in l around 0 69.8%

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

      1. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 2.7999999999999999e-282 < F < 8.9999999999999992e-208

   1. Initial program 50.0%

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

      1. associate-*l/ 50.0%

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

      2. *-lft-identity 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in l around 0 49.4%

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

      1. unpow2 49.4%

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

   6. Simplified 49.4%

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

      1. associate-/r* 49.4%

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

      2. div-inv 49.4%

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

   8. Applied egg-rr 49.4%

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

   9. Taylor expanded in F around 0 49.4%

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

       1. mul-1-neg 49.4%

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

       2. unpow2 49.4%

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

       3. associate-/l* 49.4%

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

       4. associate-*r/ 49.4%

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

   11. Simplified 49.4%

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

       1. *-un-lft-identity 49.4%

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

       2. *-commutative 49.4%

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

       3. times-frac 76.2%

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

       4. clear-num 76.2%

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

   13. Applied egg-rr 76.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-282} \lor \neg \left(F \leq 9 \cdot 10^{-208}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \]

Alternative 10: 74.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.3 \cdot 10^{-282} \lor \neg \left(F \leq 5.8 \cdot 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 3.3e-282) (not (<= F 5.8e-205)))
   (* PI l)
   (* PI (- (/ (/ l F) F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 3.3e-282) || !(F <= 5.8e-205)) {
		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 <= 3.3e-282) || !(F <= 5.8e-205)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * -((l / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 3.3e-282) or not (F <= 5.8e-205):
		tmp = math.pi * l
	else:
		tmp = math.pi * -((l / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 3.3e-282) || !(F <= 5.8e-205))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(-Float64(Float64(l / F) / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 3.3e-282) || ~((F <= 5.8e-205)))
		tmp = pi * l;
	else
		tmp = pi * -((l / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 3.3e-282], N[Not[LessEqual[F, 5.8e-205]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * (-N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.3e-282 or 5.80000000000000036e-205 < F

   1. Initial program 78.4%

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

      1. associate-*l/ 78.5%

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

      2. *-lft-identity 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in l around 0 69.8%

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

      1. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in F around inf 78.8%

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

   if 3.3e-282 < F < 5.80000000000000036e-205

   1. Initial program 50.0%

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

      1. associate-*l/ 50.0%

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

      2. *-lft-identity 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in l around 0 49.4%

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

      1. unpow2 49.4%

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

   6. Simplified 49.4%

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

      1. associate-/r* 49.4%

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

      2. div-inv 49.4%

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

   8. Applied egg-rr 49.4%

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

   9. Taylor expanded in F around 0 49.4%

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

       1. mul-1-neg 49.4%

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

       2. unpow2 49.4%

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

       3. associate-/l* 49.4%

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

       4. associate-*r/ 49.4%

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

   11. Simplified 49.4%

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

       1. associate-/r* 76.2%

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

       2. associate-/r/ 76.4%

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

   13. Applied egg-rr 76.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.3 \cdot 10^{-282} \lor \neg \left(F \leq 5.8 \cdot 10^{-205}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

Alternative 11: 74.6% 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 76.6%

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

   1. associate-*l/ 76.6%

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

   2. *-lft-identity 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in l around 0 68.4%

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

   1. unpow2 68.4%

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

6. Simplified 68.4%

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

7. Taylor expanded in F around inf 75.2%

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

8. Final simplification 75.2%

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

Reproduce

herbie shell --seed 2023242 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))