VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 98.6%

Time: 15.4s

Alternatives: 13

Speedup: 1.5×

• could not determine a ground truth

  1. F = -1.0087044146778793e-166
  2. l = -317791965457324.5

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: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 500000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+26) (not (<= (* PI l) 500000000.0)))
   (* 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) <= -1e+26) || !((((double) M_PI) * l) <= 500000000.0)) {
		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) <= -1e+26) || !((Math.PI * l) <= 500000000.0)) {
		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) <= -1e+26) or not ((math.pi * l) <= 500000000.0):
		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) <= -1e+26) || !(Float64(pi * l) <= 500000000.0))
		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) <= -1e+26) || ~(((pi * l) <= 500000000.0)))
		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], -1e+26], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000000.0]], $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) < -1.00000000000000005e26 or 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   2. Taylor expanded in l around 0 48.4%

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

      1. associate-/l* 49.2%

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

      2. unpow2 49.2%

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

   4. Simplified 49.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -1.00000000000000005e26 < (*.f64 (PI.f64) l) < 5e8

   1. Initial program 86.7%

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

      1. associate-/r* 86.7%

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

      2. metadata-eval 86.7%

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

      3. add-sqr-sqrt 45.7%

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

      4. sqrt-prod 72.4%

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

      5. sqrt-div 72.4%

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

      6. associate-*l/ 72.5%

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

      7. sqrt-div 73.5%

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

      8. metadata-eval 73.5%

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

      9. sqrt-prod 55.5%

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

      10. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 500000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 500000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+26) (not (<= (* PI l) 500000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+26) || !((((double) M_PI) * l) <= 500000000.0)) {
		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+26) || !((Math.PI * l) <= 500000000.0)) {
		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+26) or not ((math.pi * l) <= 500000000.0):
		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+26) || !(Float64(pi * l) <= 500000000.0))
		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+26) || ~(((pi * l) <= 500000000.0)))
		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+26], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000000.0]], $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) < -1.00000000000000005e26 or 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   2. Taylor expanded in l around 0 48.4%

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

      1. associate-/l* 49.2%

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

      2. unpow2 49.2%

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

   4. Simplified 49.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -1.00000000000000005e26 < (*.f64 (PI.f64) l) < 5e8

   1. Initial program 86.7%

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

      1. associate-*l/ 87.8%

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

      2. *-un-lft-identity 87.8%

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

      3. associate-/r* 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+26} \lor \neg \left(\pi \cdot \ell \leq 500000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -500000000.0) (not (<= (* PI l) 2e-16)))
   (* PI l)
   (+ (* PI l) (/ (* (* PI l) (/ -1.0 F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -500000000.0) || !((((double) M_PI) * l) <= 2e-16)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + (((((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) <= -500000000.0) || !((Math.PI * l) <= 2e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + (((Math.PI * l) * (-1.0 / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -500000000.0) or not ((math.pi * l) <= 2e-16):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + (((math.pi * l) * (-1.0 / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -500000000.0) || !(Float64(pi * l) <= 2e-16))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(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) <= -500000000.0) || ~(((pi * l) <= 2e-16)))
		tmp = pi * l;
	else
		tmp = (pi * l) + (((pi * l) * (-1.0 / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -500000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[(Pi * l), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e8 or 2e-16 < (*.f64 (PI.f64) l)

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -5e8 < (*.f64 (PI.f64) l) < 2e-16

   1. Initial program 86.6%

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

      1. associate-/r* 86.6%

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

      2. metadata-eval 86.6%

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

      3. add-sqr-sqrt 45.5%

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

      4. sqrt-prod 71.7%

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

      5. sqrt-div 71.7%

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

      6. associate-*l/ 71.8%

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

      7. sqrt-div 72.8%

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

      8. metadata-eval 72.8%

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

      9. sqrt-prod 55.7%

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

      10. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in l around 0 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \end{array} \]

Alternative 4: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -500000000.0) (not (<= (* PI l) 2e-16)))
   (* PI l)
   (- (* PI l) (/ (/ PI (/ F l)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -500000000.0) || !((((double) M_PI) * l) <= 2e-16)) {
		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) <= -500000000.0) || !((Math.PI * l) <= 2e-16)) {
		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) <= -500000000.0) or not ((math.pi * l) <= 2e-16):
		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) <= -500000000.0) || !(Float64(pi * l) <= 2e-16))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F / l)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -500000000.0) || ~(((pi * l) <= 2e-16)))
		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], -500000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F / l), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e8 or 2e-16 < (*.f64 (PI.f64) l)

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -5e8 < (*.f64 (PI.f64) l) < 2e-16

   1. Initial program 86.6%

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

      1. associate-/r* 86.6%

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

      2. metadata-eval 86.6%

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

      3. add-sqr-sqrt 45.5%

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

      4. sqrt-prod 71.7%

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

      5. sqrt-div 71.7%

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

      6. associate-*l/ 71.8%

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

      7. sqrt-div 72.8%

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

      8. metadata-eval 72.8%

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

      9. sqrt-prod 55.7%

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

      10. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in l around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/l* 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000 \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \end{array} \]

Alternative 5: 92.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 0.5 < l

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -0.5 < l < 0.5

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0 87.7%

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

      1. associate-/l* 87.6%

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

      2. unpow2 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 93.0%

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

Alternative 6: 92.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(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 <= -0.5) || !(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 <= -0.5) 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 <= -0.5) || !(l <= 0.5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * l) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -0.5) || ~((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, -0.5], N[Not[LessEqual[l, 0.5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * l), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 0.5 < l

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -0.5 < l < 0.5

   1. Initial program 86.6%

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

      1. sqr-neg 86.6%

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

      2. associate-*l/ 87.7%

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

      3. *-lft-identity 87.7%

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

      4. sqr-neg 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in l around 0 87.7%

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

4. Final simplification 93.0%

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

Alternative 7: 98.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -0.5 or 0.5 < l

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -0.5 < l < 0.5

   1. Initial program 86.6%

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

      1. associate-/r* 86.6%

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

      2. metadata-eval 86.6%

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

      3. add-sqr-sqrt 45.5%

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

      4. sqrt-prod 71.7%

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

      5. sqrt-div 71.7%

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

      6. associate-*l/ 71.8%

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

      7. sqrt-div 72.8%

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

      8. metadata-eval 72.8%

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

      9. sqrt-prod 55.7%

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

      10. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in l around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 98.9%

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

Alternative 8: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-106} \lor \neg \left(F \leq 6 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;{F}^{-2} \cdot \left(\pi \cdot \left(-\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 2.8e-265)
   (/ (- l) (* F (/ F PI)))
   (if (or (<= F 3.1e-106) (not (<= F 6e-73)))
     (* PI l)
     (* (pow F -2.0) (* PI (- l))))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 2.8e-265) {
		tmp = -l / (F * (F / ((double) M_PI)));
	} else if ((F <= 3.1e-106) || !(F <= 6e-73)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = pow(F, -2.0) * (((double) M_PI) * -l);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 2.8e-265) {
		tmp = -l / (F * (F / Math.PI));
	} else if ((F <= 3.1e-106) || !(F <= 6e-73)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.pow(F, -2.0) * (Math.PI * -l);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 2.8e-265:
		tmp = -l / (F * (F / math.pi))
	elif (F <= 3.1e-106) or not (F <= 6e-73):
		tmp = math.pi * l
	else:
		tmp = math.pow(F, -2.0) * (math.pi * -l)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 2.8e-265)
		tmp = Float64(Float64(-l) / Float64(F * Float64(F / pi)));
	elseif ((F <= 3.1e-106) || !(F <= 6e-73))
		tmp = Float64(pi * l);
	else
		tmp = Float64((F ^ -2.0) * Float64(pi * Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 2.8e-265)
		tmp = -l / (F * (F / pi));
	elseif ((F <= 3.1e-106) || ~((F <= 6e-73)))
		tmp = pi * l;
	else
		tmp = (F ^ -2.0) * (pi * -l);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 2.8e-265], N[((-l) / N[(F * N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3.1e-106], N[Not[LessEqual[F, 6e-73]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[Power[F, -2.0], $MachinePrecision] * N[(Pi * (-l)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.80000000000000023e-265

   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 67.4%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in F around 0 21.9%

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

      1. mul-1-neg 21.9%

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

      2. unpow2 21.9%

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

      3. associate-/l* 21.9%

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

      4. associate-*r/ 21.9%

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

   7. Simplified 21.9%

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

   if 2.80000000000000023e-265 < F < 3.09999999999999985e-106 or 6e-73 < F

   1. Initial program 72.9%

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

   2. Taylor expanded in l around 0 69.4%

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

      1. associate-/l* 69.3%

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

      2. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in F around inf 75.2%

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

   if 3.09999999999999985e-106 < F < 6e-73

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 50.1%

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

      1. associate-/l* 49.7%

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

      2. unpow2 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in F around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. unpow2 49.7%

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

      3. *-commutative 49.7%

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

      4. *-lft-identity 49.7%

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

      5. *-commutative 49.7%

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

      6. associate-*r/ 49.7%

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

      7. associate-/r* 49.7%

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

      8. *-rgt-identity 49.7%

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

      9. associate-*r/ 49.7%

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

      10. unpow-1 49.7%

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

      11. unpow-1 49.7%

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

      12. pow-sqr 50.1%

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

      13. metadata-eval 50.1%

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

      14. distribute-rgt-neg-in 50.1%

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

      15. *-commutative 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-106} \lor \neg \left(F \leq 6 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;{F}^{-2} \cdot \left(\pi \cdot \left(-\ell\right)\right)\\ \end{array} \]

Alternative 9: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-106} \lor \neg \left(F \leq 5 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{{F}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 2.9e-265)
   (/ (- l) (* F (/ F PI)))
   (if (or (<= F 2.6e-106) (not (<= F 5e-73)))
     (* PI l)
     (/ (* PI (- l)) (pow F 2.0)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 2.9e-265) {
		tmp = -l / (F * (F / ((double) M_PI)));
	} else if ((F <= 2.6e-106) || !(F <= 5e-73)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * -l) / pow(F, 2.0);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 2.9e-265) {
		tmp = -l / (F * (F / Math.PI));
	} else if ((F <= 2.6e-106) || !(F <= 5e-73)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * -l) / Math.pow(F, 2.0);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 2.9e-265:
		tmp = -l / (F * (F / math.pi))
	elif (F <= 2.6e-106) or not (F <= 5e-73):
		tmp = math.pi * l
	else:
		tmp = (math.pi * -l) / math.pow(F, 2.0)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 2.9e-265)
		tmp = Float64(Float64(-l) / Float64(F * Float64(F / pi)));
	elseif ((F <= 2.6e-106) || !(F <= 5e-73))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * Float64(-l)) / (F ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 2.9e-265)
		tmp = -l / (F * (F / pi));
	elseif ((F <= 2.6e-106) || ~((F <= 5e-73)))
		tmp = pi * l;
	else
		tmp = (pi * -l) / (F ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 2.9e-265], N[((-l) / N[(F * N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 2.6e-106], N[Not[LessEqual[F, 5e-73]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * (-l)), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.89999999999999975e-265

   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 67.4%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in F around 0 21.9%

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

      1. mul-1-neg 21.9%

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

      2. unpow2 21.9%

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

      3. associate-/l* 21.9%

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

      4. associate-*r/ 21.9%

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

   7. Simplified 21.9%

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

   if 2.89999999999999975e-265 < F < 2.6000000000000001e-106 or 4.9999999999999998e-73 < F

   1. Initial program 72.9%

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

   2. Taylor expanded in l around 0 69.4%

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

      1. associate-/l* 69.3%

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

      2. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in F around inf 75.2%

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

   if 2.6000000000000001e-106 < F < 4.9999999999999998e-73

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 50.1%

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

      1. associate-/l* 49.7%

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

      2. unpow2 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in F around 0 50.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-106} \lor \neg \left(F \leq 5 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{{F}^{2}}\\ \end{array} \]

Alternative 10: 92.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double F, double l) {
	double tmp;
	if ((l <= -0.5) || !(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 <= -0.5) || !(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 <= -0.5) 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 <= -0.5) || !(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 <= -0.5) || ~((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, -0.5], 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 < -0.5 or 0.5 < l

   1. Initial program 58.6%

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

   2. Taylor expanded in l around 0 48.8%

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

      1. associate-/l* 49.6%

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

      2. unpow2 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in F around inf 98.2%

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

   if -0.5 < l < 0.5

   1. Initial program 86.6%

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

   2. Taylor expanded in l around 0 87.7%

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

      1. associate-/l* 87.6%

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

      2. unpow2 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in l around 0 86.6%

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

      1. sub-neg 86.6%

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

      2. mul-1-neg 86.6%

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

      3. distribute-rgt-in 86.6%

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

      4. mul-1-neg 86.6%

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

      5. *-lft-identity 86.6%

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

      6. unpow2 86.6%

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

      7. associate-*l/ 86.6%

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

      8. associate-/r* 86.6%

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

      9. *-rgt-identity 86.6%

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

      10. associate-*r/ 86.5%

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

      11. unpow-1 86.5%

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

      12. unpow-1 86.5%

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

      13. pow-sqr 86.6%

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

      14. metadata-eval 86.6%

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

      15. *-commutative 86.6%

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

      16. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 92.5%

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

Alternative 11: 47.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.2 \cdot 10^{-265} \lor \neg \left(F \leq 2.5 \cdot 10^{-106}\right) \land F \leq 7.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= F 2.2e-265) (and (not (<= F 2.5e-106)) (<= F 7.5e-73)))
   (/ (- l) (* F (/ F PI)))
   (* PI l)))

C

double code(double F, double l) {
	double tmp;
	if ((F <= 2.2e-265) || (!(F <= 2.5e-106) && (F <= 7.5e-73))) {
		tmp = -l / (F * (F / ((double) M_PI)));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F <= 2.2e-265) || (!(F <= 2.5e-106) && (F <= 7.5e-73))) {
		tmp = -l / (F * (F / Math.PI));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F <= 2.2e-265) or (not (F <= 2.5e-106) and (F <= 7.5e-73)):
		tmp = -l / (F * (F / math.pi))
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((F <= 2.2e-265) || (!(F <= 2.5e-106) && (F <= 7.5e-73)))
		tmp = Float64(Float64(-l) / Float64(F * Float64(F / pi)));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F <= 2.2e-265) || (~((F <= 2.5e-106)) && (F <= 7.5e-73)))
		tmp = -l / (F * (F / pi));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[F, 2.2e-265], And[N[Not[LessEqual[F, 2.5e-106]], $MachinePrecision], LessEqual[F, 7.5e-73]]], N[((-l) / N[(F * N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.2000000000000001e-265 or 2.49999999999999991e-106 < F < 7.5e-73

   1. Initial program 72.1%

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

   2. Taylor expanded in l around 0 66.9%

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

      1. associate-/l* 67.6%

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

      2. unpow2 67.6%

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

   4. Simplified 67.6%

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

   5. Taylor expanded in F around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unpow2 22.7%

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

      3. associate-/l* 22.7%

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

      4. associate-*r/ 22.7%

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

   7. Simplified 22.7%

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

   if 2.2000000000000001e-265 < F < 2.49999999999999991e-106 or 7.5e-73 < F

   1. Initial program 72.9%

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

   2. Taylor expanded in l around 0 69.4%

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

      1. associate-/l* 69.3%

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

      2. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in F around inf 75.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.2 \cdot 10^{-265} \lor \neg \left(F \leq 2.5 \cdot 10^{-106}\right) \land F \leq 7.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 12: 47.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-106} \lor \neg \left(F \leq 4.4 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F}{\frac{\pi}{F}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.9e-265)
   (/ (- l) (* F (/ F PI)))
   (if (or (<= F 1.85e-106) (not (<= F 4.4e-73)))
     (* PI l)
     (/ (- l) (/ F (/ PI F))))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.9e-265) {
		tmp = -l / (F * (F / ((double) M_PI)));
	} else if ((F <= 1.85e-106) || !(F <= 4.4e-73)) {
		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 <= 1.9e-265) {
		tmp = -l / (F * (F / Math.PI));
	} else if ((F <= 1.85e-106) || !(F <= 4.4e-73)) {
		tmp = Math.PI * l;
	} else {
		tmp = -l / (F / (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.9e-265:
		tmp = -l / (F * (F / math.pi))
	elif (F <= 1.85e-106) or not (F <= 4.4e-73):
		tmp = math.pi * l
	else:
		tmp = -l / (F / (math.pi / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.9e-265)
		tmp = Float64(Float64(-l) / Float64(F * Float64(F / pi)));
	elseif ((F <= 1.85e-106) || !(F <= 4.4e-73))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-l) / Float64(F / Float64(pi / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.9e-265)
		tmp = -l / (F * (F / pi));
	elseif ((F <= 1.85e-106) || ~((F <= 4.4e-73)))
		tmp = pi * l;
	else
		tmp = -l / (F / (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.9e-265], N[((-l) / N[(F * N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.85e-106], N[Not[LessEqual[F, 4.4e-73]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[((-l) / N[(F / N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.8999999999999999e-265

   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 67.4%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in F around 0 21.9%

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

      1. mul-1-neg 21.9%

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

      2. unpow2 21.9%

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

      3. associate-/l* 21.9%

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

      4. associate-*r/ 21.9%

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

   7. Simplified 21.9%

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

   if 1.8999999999999999e-265 < F < 1.8499999999999999e-106 or 4.4e-73 < F

   1. Initial program 72.9%

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

   2. Taylor expanded in l around 0 69.4%

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

      1. associate-/l* 69.3%

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

      2. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in F around inf 75.2%

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

   if 1.8499999999999999e-106 < F < 4.4e-73

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 50.1%

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

      1. associate-/l* 49.7%

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

      2. unpow2 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in F around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. unpow2 49.7%

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

      3. associate-/l* 49.7%

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

      4. associate-*r/ 49.7%

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

   7. Simplified 49.7%

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

      1. clear-num 49.7%

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

      2. un-div-inv 49.7%

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

   9. Applied egg-rr 49.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-106} \lor \neg \left(F \leq 4.4 \cdot 10^{-73}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\frac{F}{\frac{\pi}{F}}}\\ \end{array} \]

Alternative 13: 73.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in l around 0 68.1%

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

   1. associate-/l* 68.5%

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

   2. unpow2 68.5%

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

4. Simplified 68.5%

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

5. Taylor expanded in F around inf 74.5%

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

6. Final simplification 74.5%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))