VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 99.0%

Time: 15.4s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. F = -9.557577841980218e-252
  2. l = 3548295278916407.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+21) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (+ (* PI l) (/ (* (tan (* PI l)) (/ -1.0 F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+21) || !((((double) M_PI) * l) <= 500000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) * (-1.0 / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+21) || !((Math.PI * l) <= 500000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) * (-1.0 / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+21) or not ((math.pi * l) <= 500000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) * (-1.0 / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+21) || !(Float64(pi * l) <= 500000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) * Float64(-1.0 / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+21) || ~(((pi * l) <= 500000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) * (-1.0 / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+21], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

   2. Taylor expanded in l around 0 51.4%

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

      1. associate-/l* 51.4%

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

      2. associate-/r/ 51.4%

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

      3. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.4%

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

      4. associate-/l* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 85.9%

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

      1. associate-/r* 85.9%

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

      2. metadata-eval 85.9%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-prod 67.8%

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

      5. sqrt-div 67.8%

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

      6. associate-*l/ 67.8%

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

      7. sqrt-div 69.8%

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

      8. metadata-eval 69.8%

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

      9. sqrt-prod 56.8%

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

      10. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+21) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+21) || !((((double) M_PI) * l) <= 500000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+21) || !((Math.PI * l) <= 500000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+21) or not ((math.pi * l) <= 500000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+21) || !(Float64(pi * l) <= 500000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+21) || ~(((pi * l) <= 500000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+21], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $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) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

   2. Taylor expanded in l around 0 51.4%

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

      1. associate-/l* 51.4%

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

      2. associate-/r/ 51.4%

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

      3. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.4%

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

      4. associate-/l* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 85.9%

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

      1. associate-*l/ 87.9%

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

      2. *-un-lft-identity 87.9%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+21) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+21) || !((((double) M_PI) * l) <= 500000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+21) || !((Math.PI * l) <= 500000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+21) or not ((math.pi * l) <= 500000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+21) || !(Float64(pi * l) <= 500000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+21) || ~(((pi * l) <= 500000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+21], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

   2. Taylor expanded in l around 0 51.4%

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

      1. associate-/l* 51.4%

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

      2. associate-/r/ 51.4%

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

      3. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.4%

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

      4. associate-/l* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0 86.6%

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

      1. associate-/l* 86.6%

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

      2. associate-/r/ 86.6%

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

      3. unpow2 86.6%

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

   4. Simplified 86.6%

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

   5. Taylor expanded in l around 0 86.6%

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

      1. unpow2 86.6%

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

      2. *-commutative 86.6%

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

      3. times-frac 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 4: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+21) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (- (* PI l) (/ PI (* F (/ F l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+21) || !((((double) M_PI) * l) <= 500000.0)) {
		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) <= -2e+21) || !((Math.PI * l) <= 500000.0)) {
		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) <= -2e+21) or not ((math.pi * l) <= 500000.0):
		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) <= -2e+21) || !(Float64(pi * l) <= 500000.0))
		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) <= -2e+21) || ~(((pi * l) <= 500000.0)))
		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], -2e+21], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $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) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

   2. Taylor expanded in l around 0 51.4%

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

      1. associate-/l* 51.4%

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

      2. associate-/r/ 51.4%

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

      3. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.4%

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

      4. associate-/l* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0 86.6%

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

      1. associate-/l* 86.6%

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

      2. associate-/r/ 86.6%

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

      3. unpow2 86.6%

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

   4. Simplified 86.6%

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

      1. *-commutative 86.6%

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

      2. clear-num 86.6%

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

      3. un-div-inv 86.6%

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

      4. associate-/l* 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-/r/ 97.9%

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+21) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (* PI (- l (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+21) || !((((double) M_PI) * l) <= 500000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (l - (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+21) || !((Math.PI * l) <= 500000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (l - (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+21) or not ((math.pi * l) <= 500000.0):
		tmp = math.pi * l
	else:
		tmp = math.pi * (l - (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+21) || !(Float64(pi * l) <= 500000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(l - Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+21) || ~(((pi * l) <= 500000.0)))
		tmp = pi * l;
	else
		tmp = pi * (l - (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+21], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(l - N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

   2. Taylor expanded in l around 0 51.4%

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

      1. associate-/l* 51.4%

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

      2. associate-/r/ 51.4%

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

      3. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.4%

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

      4. associate-/l* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0 86.6%

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

      1. associate-/l* 86.6%

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

      2. associate-/r/ 86.6%

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

      3. unpow2 86.6%

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

   4. Simplified 86.6%

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

      1. *-commutative 86.6%

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

      2. clear-num 86.6%

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

      3. un-div-inv 86.6%

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

      4. associate-/l* 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in F around 0 86.6%

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

      1. mul-1-neg 86.6%

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

      2. associate-*l/ 86.6%

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

      3. unpow2 86.6%

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

      4. distribute-lft-neg-in 86.6%

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

      5. distribute-rgt-in 86.6%

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

      6. sub-neg 86.6%

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

   9. Simplified 86.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \]

Alternative 6: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{-129} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-295}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e-129) (not (<= (* PI l) 5e-295)))
   (* PI l)
   (* PI (/ (- l) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e-129) || !((((double) M_PI) * l) <= 5e-295)) {
		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 (((Math.PI * l) <= -1e-129) || !((Math.PI * l) <= 5e-295)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (-l / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -1e-129) or not ((math.pi * l) <= 5e-295):
		tmp = math.pi * l
	else:
		tmp = math.pi * (-l / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e-129) || !(Float64(pi * l) <= 5e-295))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(Float64(-l) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -1e-129) || ~(((pi * l) <= 5e-295)))
		tmp = pi * l;
	else
		tmp = pi * (-l / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e-129], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-295]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[((-l) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -9.9999999999999993e-130 or 5.00000000000000008e-295 < (*.f64 (PI.f64) l)

   1. Initial program 72.0%

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

   2. Taylor expanded in l around 0 65.6%

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

      1. associate-/l* 65.6%

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

      2. associate-/r/ 65.6%

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

      3. unpow2 65.6%

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

   4. Simplified 65.6%

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

      1. *-commutative 65.6%

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

      2. clear-num 65.6%

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

      3. un-div-inv 65.6%

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

      4. associate-/l* 69.4%

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

   6. Applied egg-rr 69.4%

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

   7. Taylor expanded in F around inf 83.5%

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

   if -9.9999999999999993e-130 < (*.f64 (PI.f64) l) < 5.00000000000000008e-295

   1. Initial program 83.4%

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

   2. Taylor expanded in l around 0 83.4%

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

      1. associate-/l* 83.5%

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

      2. associate-/r/ 83.4%

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

      3. unpow2 83.4%

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

   4. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. clear-num 83.2%

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

      3. un-div-inv 83.3%

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

      4. associate-/l* 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in F around 0 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*r/ 60.1%

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

      3. neg-mul-1 60.1%

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

      4. distribute-rgt-neg-out 60.1%

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

      5. associate-*r/ 60.1%

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

      6. unpow2 60.1%

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

   9. Simplified 60.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{-129} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-295}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \end{array} \]

Alternative 7: 74.1% 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 73.3%

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

2. Taylor expanded in l around 0 67.6%

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

   1. associate-/l* 67.6%

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

   2. associate-/r/ 67.6%

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

   3. unpow2 67.6%

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

4. Simplified 67.6%

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

   1. *-commutative 67.6%

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

   2. clear-num 67.6%

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

   3. un-div-inv 67.6%

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

   4. associate-/l* 72.8%

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

6. Applied egg-rr 72.8%

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

7. Taylor expanded in F around inf 77.0%

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

8. Final simplification 77.0%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))