VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.7% → 99.0%

Time: 12.2s

Alternatives: 9

Speedup: 1.5×

• could not determine a ground truth

  1. F = 2.9163409770289628e+178
  2. l = -4370274241733294.5

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: 76.7% 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: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 40000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+17) (not (<= (* PI l) 40000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+17) || !((((double) M_PI) * l) <= 40000000000.0)) {
		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) <= -1e+17) || !((Math.PI * l) <= 40000000000.0)) {
		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) <= -1e+17) or not ((math.pi * l) <= 40000000000.0):
		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) <= -1e+17) || !(Float64(pi * l) <= 40000000000.0))
		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) <= -1e+17) || ~(((pi * l) <= 40000000000.0)))
		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], -1e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 40000000000.0]], $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) < -1e17 or 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.3%

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

      1. associate-*l/ 62.3%

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

      2. *-lft-identity 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 99.7%

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

   if -1e17 < (*.f64 (PI.f64) l) < 4e10

   1. Initial program 85.5%

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

      1. associate-*l/ 86.1%

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

      2. *-un-lft-identity 86.1%

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

      3. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 40000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200000000000 \lor \neg \left(\pi \cdot \ell \leq 40000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -200000000000.0) (not (<= (* PI l) 40000000000.0)))
   (* PI l)
   (- (* PI l) (/ PI (* F (/ F l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -200000000000.0) || !((((double) M_PI) * l) <= 40000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F * (F / l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -200000000000.0) || !((Math.PI * l) <= 40000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI / (F * (F / l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -200000000000.0) or not ((math.pi * l) <= 40000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi / (F * (F / l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -200000000000.0) || !(Float64(pi * l) <= 40000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F * Float64(F / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -200000000000.0) || ~(((pi * l) <= 40000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi / (F * (F / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -200000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 40000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e11 or 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -2e11 < (*.f64 (PI.f64) l) < 4e10

   1. Initial program 85.3%

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

      1. associate-*l/ 85.9%

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

      2. *-un-lft-identity 85.9%

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

      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 85.9%

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

      1. *-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. times-frac 99.7%

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

   6. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. frac-times 99.7%

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

      4. *-un-lft-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -200000000000 \lor \neg \left(\pi \cdot \ell \leq 40000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 3: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi + \frac{\pi}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -17500000000.0) (not (<= l 67000000000.0)))
   (* PI l)
   (* l (+ PI (* (/ PI F) (/ -1.0 F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) + ((((double) M_PI) / F) * (-1.0 / F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI + ((Math.PI / F) * (-1.0 / F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -17500000000.0) or not (l <= 67000000000.0):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi + ((math.pi / F) * (-1.0 / F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -17500000000.0) || !(l <= 67000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi + Float64(Float64(pi / F) * Float64(-1.0 / F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -17500000000.0) || ~((l <= 67000000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi + ((pi / F) * (-1.0 / F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -17500000000.0], N[Not[LessEqual[l, 67000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi + N[(N[(Pi / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.75e10 or 6.7e10 < l

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -1.75e10 < l < 6.7e10

   1. Initial program 85.3%

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

      1. associate-*l/ 85.9%

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

      2. *-lft-identity 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in l around 0 85.3%

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

      1. unpow2 85.3%

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

   6. Simplified 85.3%

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

      1. associate-/r* 85.4%

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

      2. div-inv 85.3%

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

   8. Applied egg-rr 85.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi + \frac{\pi}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \]

Alternative 4: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -17500000000.0) (not (<= l 67000000000.0)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		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 <= -17500000000.0) || !(l <= 67000000000.0)) {
		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 <= -17500000000.0) or not (l <= 67000000000.0):
		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 <= -17500000000.0) || !(l <= 67000000000.0))
		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 <= -17500000000.0) || ~((l <= 67000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -17500000000.0], N[Not[LessEqual[l, 67000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.75e10 or 6.7e10 < l

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -1.75e10 < l < 6.7e10

   1. Initial program 85.3%

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

   2. Taylor expanded in l around 0 85.9%

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

      1. associate-/l* 85.9%

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

      2. associate-/r/ 85.9%

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

      3. unpow2 85.9%

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

   4. Simplified 85.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -17500000000.0) (not (<= l 67000000000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -17500000000.0) or not (l <= 67000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -17500000000.0) || !(l <= 67000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -17500000000.0) || ~((l <= 67000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -17500000000.0], N[Not[LessEqual[l, 67000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.75e10 or 6.7e10 < l

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -1.75e10 < l < 6.7e10

   1. Initial program 85.3%

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

      1. associate-*l/ 85.9%

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

      2. *-un-lft-identity 85.9%

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

      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 85.9%

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

      1. *-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. times-frac 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 6: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -17500000000.0) (not (<= l 67000000000.0)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -17500000000.0) || !(l <= 67000000000.0)) {
		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 <= -17500000000.0) || !(l <= 67000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -17500000000.0) or not (l <= 67000000000.0):
		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 <= -17500000000.0) || !(l <= 67000000000.0))
		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 <= -17500000000.0) || ~((l <= 67000000000.0)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -17500000000.0], N[Not[LessEqual[l, 67000000000.0]], $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 < -1.75e10 or 6.7e10 < l

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around 0 48.0%

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

      1. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in F around inf 98.9%

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

   if -1.75e10 < l < 6.7e10

   1. Initial program 85.3%

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

      1. associate-*l/ 85.9%

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

      2. *-lft-identity 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in l around 0 85.3%

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

      1. unpow2 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -17500000000 \lor \neg \left(\ell \leq 67000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 73.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-5} \lor \neg \left(\ell \leq -1.55 \cdot 10^{-109}\right) \land \left(\ell \leq 2.6 \cdot 10^{-126} \lor \neg \left(\ell \leq 3 \cdot 10^{-50}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1e-5)
         (and (not (<= l -1.55e-109)) (or (<= l 2.6e-126) (not (<= l 3e-50)))))
   (* PI l)
   (* l (/ (- PI) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1e-5) || (!(l <= -1.55e-109) && ((l <= 2.6e-126) || !(l <= 3e-50)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (-((double) M_PI) / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1e-5) || (!(l <= -1.55e-109) && ((l <= 2.6e-126) || !(l <= 3e-50)))) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (-Math.PI / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1e-5) or (not (l <= -1.55e-109) and ((l <= 2.6e-126) or not (l <= 3e-50))):
		tmp = math.pi * l
	else:
		tmp = l * (-math.pi / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1e-5) || (!(l <= -1.55e-109) && ((l <= 2.6e-126) || !(l <= 3e-50))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(Float64(-pi) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1e-5) || (~((l <= -1.55e-109)) && ((l <= 2.6e-126) || ~((l <= 3e-50)))))
		tmp = pi * l;
	else
		tmp = l * (-pi / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1e-5], And[N[Not[LessEqual[l, -1.55e-109]], $MachinePrecision], Or[LessEqual[l, 2.6e-126], N[Not[LessEqual[l, 3e-50]], $MachinePrecision]]]], N[(Pi * l), $MachinePrecision], N[(l * N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.00000000000000008e-5 or -1.55e-109 < l < 2.59999999999999999e-126 or 2.9999999999999999e-50 < l

   1. Initial program 70.2%

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

      1. associate-*l/ 70.6%

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

      2. *-lft-identity 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in l around 0 61.8%

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

      1. unpow2 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in F around inf 78.3%

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

   if -1.00000000000000008e-5 < l < -1.55e-109 or 2.59999999999999999e-126 < l < 2.9999999999999999e-50

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

      3. unpow2 70.2%

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

   9. Simplified 70.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-5} \lor \neg \left(\ell \leq -1.55 \cdot 10^{-109}\right) \land \left(\ell \leq 2.6 \cdot 10^{-126} \lor \neg \left(\ell \leq 3 \cdot 10^{-50}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \end{array} \]

Alternative 8: 73.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-9} \lor \neg \left(\ell \leq -1.5 \cdot 10^{-109} \lor \neg \left(\ell \leq 1.5 \cdot 10^{-127}\right) \land \ell \leq 2.05 \cdot 10^{-50}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{F}}{-F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -7e-9)
         (not
          (or (<= l -1.5e-109) (and (not (<= l 1.5e-127)) (<= l 2.05e-50)))))
   (* PI l)
   (* l (/ (/ PI F) (- F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -7e-9) || !((l <= -1.5e-109) || (!(l <= 1.5e-127) && (l <= 2.05e-50)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * ((((double) M_PI) / F) / -F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -7e-9) || !((l <= -1.5e-109) || (!(l <= 1.5e-127) && (l <= 2.05e-50)))) {
		tmp = Math.PI * l;
	} else {
		tmp = l * ((Math.PI / F) / -F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -7e-9) or not ((l <= -1.5e-109) or (not (l <= 1.5e-127) and (l <= 2.05e-50))):
		tmp = math.pi * l
	else:
		tmp = l * ((math.pi / F) / -F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -7e-9) || !((l <= -1.5e-109) || (!(l <= 1.5e-127) && (l <= 2.05e-50))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(Float64(pi / F) / Float64(-F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -7e-9) || ~(((l <= -1.5e-109) || (~((l <= 1.5e-127)) && (l <= 2.05e-50)))))
		tmp = pi * l;
	else
		tmp = l * ((pi / F) / -F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -7e-9], N[Not[Or[LessEqual[l, -1.5e-109], And[N[Not[LessEqual[l, 1.5e-127]], $MachinePrecision], LessEqual[l, 2.05e-50]]]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(N[(Pi / F), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.9999999999999998e-9 or -1.50000000000000011e-109 < l < 1.50000000000000004e-127 or 2.04999999999999993e-50 < l

   1. Initial program 70.2%

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

      1. associate-*l/ 70.6%

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

      2. *-lft-identity 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in l around 0 61.8%

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

      1. unpow2 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in F around inf 78.3%

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

   if -6.9999999999999998e-9 < l < -1.50000000000000011e-109 or 1.50000000000000004e-127 < l < 2.04999999999999993e-50

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 70.2%

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

      1. mul-1-neg 70.2%

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

      2. unpow2 70.2%

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

      3. associate-/r* 70.4%

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

      4. *-lft-identity 70.4%

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

      5. associate-/l* 70.3%

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

      6. associate-/l* 70.2%

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

      7. unpow2 70.2%

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

      8. associate-/r/ 70.2%

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

      9. /-rgt-identity 70.2%

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

      10. unpow2 70.2%

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

      11. associate-/l* 70.2%

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

      12. associate-/r/ 70.1%

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

      13. unpow-1 70.1%

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

      14. unpow-1 70.1%

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

      15. pow-sqr 70.3%

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

      16. metadata-eval 70.3%

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

      17. distribute-rgt-neg-in 70.3%

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

   9. Simplified 70.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 1.7%

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

       3. sqr-neg 1.7%

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

       4. sqrt-unprod 1.7%

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

       5. add-sqr-sqrt 1.7%

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

       6. metadata-eval 1.7%

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

       7. pow-flip 1.7%

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

       8. pow2 1.7%

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

       9. associate-/r/ 1.7%

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

       10. associate-/l* 1.7%

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

       11. clear-num 1.7%

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

       12. frac-2neg 1.7%

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

       13. frac-2neg 1.7%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 70.4%

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

       16. sqr-neg 70.4%

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

       17. sqrt-unprod 69.8%

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

       18. add-sqr-sqrt 70.4%

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

       19. distribute-frac-neg 70.4%

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

       20. frac-2neg 70.4%

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

   11. Applied egg-rr 70.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-9} \lor \neg \left(\ell \leq -1.5 \cdot 10^{-109} \lor \neg \left(\ell \leq 1.5 \cdot 10^{-127}\right) \land \ell \leq 2.05 \cdot 10^{-50}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{F}}{-F}\\ \end{array} \]

Alternative 9: 74.4% 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 74.4%

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

   1. associate-*l/ 74.7%

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

   2. *-lft-identity 74.7%

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

3. Simplified 74.7%

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

4. Taylor expanded in l around 0 67.1%

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

   1. unpow2 67.1%

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

6. Simplified 67.1%

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

7. Taylor expanded in F around inf 71.7%

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

8. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))