VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.4%

Time: 13.1s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. F = -1.2485369828581167e+273
  2. l = -53830678371042.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+16) (not (<= (* PI l) 1000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+16) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+16) || !((Math.PI * l) <= 1000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+16) or not ((math.pi * l) <= 1000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+16) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+16) || ~(((pi * l) <= 1000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e16 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 64.7%

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

      1. sqr-neg 64.7%

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

      2. associate-*l/ 64.7%

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

      3. *-lft-identity 64.7%

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

      4. sqr-neg 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in l around 0 45.4%

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

      1. unpow2 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e16 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 91.3%

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

      1. associate-*l/ 91.3%

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

      2. *-un-lft-identity 91.3%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000000 \lor \neg \left(\pi \cdot \ell \leq 10^{-26}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -500000000000.0) (not (<= (* PI l) 1e-26)))
   (* PI l)
   (* PI (- l (/ (/ l F) F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -500000000000.0) || !((((double) M_PI) * l) <= 1e-26)) {
		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) <= -500000000000.0) || !((Math.PI * l) <= 1e-26)) {
		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) <= -500000000000.0) or not ((math.pi * l) <= 1e-26):
		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) <= -500000000000.0) || !(Float64(pi * l) <= 1e-26))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(l - Float64(Float64(l / F) / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -500000000000.0) || ~(((pi * l) <= 1e-26)))
		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], -500000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-26]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(l - N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e11 or 1e-26 < (*.f64 (PI.f64) l)

   1. Initial program 66.1%

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

      1. sqr-neg 66.1%

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

      2. associate-*l/ 66.1%

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

      3. *-lft-identity 66.1%

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

      4. sqr-neg 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in l around 0 45.3%

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

      1. unpow2 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in F around inf 96.7%

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

   if -5e11 < (*.f64 (PI.f64) l) < 1e-26

   1. Initial program 91.3%

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

   2. Taylor expanded in l around 0 90.7%

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

      1. associate-/l* 90.7%

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

      2. associate-/r/ 90.7%

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

      3. unpow2 90.7%

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

   4. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. distribute-lft-out-- 90.7%

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

      3. associate-/r* 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -500000000000 \lor \neg \left(\pi \cdot \ell \leq 10^{-26}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

Alternative 3: 74.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-98}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 5e-155)
   (+ (* PI l) -1.0)
   (if (<= (* F F) 1e-98) (/ (- l) (/ (* F F) PI)) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 5e-155) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if ((F * F) <= 1e-98) {
		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 * F) <= 5e-155) {
		tmp = (Math.PI * l) + -1.0;
	} else if ((F * F) <= 1e-98) {
		tmp = -l / ((F * F) / Math.PI);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 5e-155:
		tmp = (math.pi * l) + -1.0
	elif (F * F) <= 1e-98:
		tmp = -l / ((F * F) / math.pi)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 5e-155)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (Float64(F * F) <= 1e-98)
		tmp = Float64(Float64(-l) / Float64(Float64(F * F) / pi));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 5e-155)
		tmp = (pi * l) + -1.0;
	elseif ((F * F) <= 1e-98)
		tmp = -l / ((F * F) / pi);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 5e-155], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-98], N[((-l) / N[(N[(F * F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 4.9999999999999999e-155

   1. Initial program 43.5%

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

      1. associate-/r/ 43.5%

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

      2. inv-pow 43.5%

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

      3. pow-to-exp 21.8%

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

      4. associate-/l* 26.3%

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

   3. Applied egg-rr 26.3%

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

      1. exp-to-pow 54.9%

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

      2. unpow-1 54.9%

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

      3. clear-num 54.9%

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

      4. associate-/l/ 43.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. frac-times 26.6%

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

      7. div-inv 26.7%

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

      8. associate-*l* 23.6%

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

      9. add-sqr-sqrt 9.7%

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

      10. sqrt-unprod 11.1%

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

      11. frac-times 10.1%

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

      12. add-sqr-sqrt 10.1%

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

      13. associate-/l/ 11.1%

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

      14. clear-num 11.1%

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

   5. Applied egg-rr 23.6%

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

      1. *-commutative 23.6%

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

      2. associate-*l/ 23.6%

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

      3. *-lft-identity 23.6%

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

      4. associate-/r/ 23.6%

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

      5. *-inverses 54.0%

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

   7. Simplified 54.0%

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

   if 4.9999999999999999e-155 < (*.f64 F F) < 9.99999999999999939e-99

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. associate-*l/ 85.4%

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

      3. *-lft-identity 85.4%

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

      4. sqr-neg 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in l around 0 76.6%

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

      1. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in F around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. associate-/l* 76.6%

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

      3. unpow2 76.6%

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

   9. Simplified 76.6%

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

   if 9.99999999999999939e-99 < (*.f64 F F)

   1. Initial program 97.7%

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

      1. sqr-neg 97.7%

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

      2. associate-*l/ 97.7%

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

      3. *-lft-identity 97.7%

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

      4. sqr-neg 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in l around 0 86.8%

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

      1. unpow2 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in F around inf 91.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-98}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4: 74.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-98}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 5e-155)
   (+ (* PI l) -1.0)
   (if (<= (* F F) 1e-98) (/ (* PI (- l)) (* F F)) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 5e-155) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if ((F * F) <= 1e-98) {
		tmp = (((double) M_PI) * -l) / (F * F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 5e-155) {
		tmp = (Math.PI * l) + -1.0;
	} else if ((F * F) <= 1e-98) {
		tmp = (Math.PI * -l) / (F * F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 5e-155:
		tmp = (math.pi * l) + -1.0
	elif (F * F) <= 1e-98:
		tmp = (math.pi * -l) / (F * F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 5e-155)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (Float64(F * F) <= 1e-98)
		tmp = Float64(Float64(pi * Float64(-l)) / Float64(F * F));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 5e-155)
		tmp = (pi * l) + -1.0;
	elseif ((F * F) <= 1e-98)
		tmp = (pi * -l) / (F * F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 5e-155], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-98], N[(N[(Pi * (-l)), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 4.9999999999999999e-155

   1. Initial program 43.5%

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

      1. associate-/r/ 43.5%

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

      2. inv-pow 43.5%

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

      3. pow-to-exp 21.8%

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

      4. associate-/l* 26.3%

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

   3. Applied egg-rr 26.3%

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

      1. exp-to-pow 54.9%

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

      2. unpow-1 54.9%

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

      3. clear-num 54.9%

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

      4. associate-/l/ 43.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. frac-times 26.6%

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

      7. div-inv 26.7%

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

      8. associate-*l* 23.6%

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

      9. add-sqr-sqrt 9.7%

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

      10. sqrt-unprod 11.1%

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

      11. frac-times 10.1%

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

      12. add-sqr-sqrt 10.1%

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

      13. associate-/l/ 11.1%

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

      14. clear-num 11.1%

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

   5. Applied egg-rr 23.6%

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

      1. *-commutative 23.6%

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

      2. associate-*l/ 23.6%

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

      3. *-lft-identity 23.6%

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

      4. associate-/r/ 23.6%

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

      5. *-inverses 54.0%

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

   7. Simplified 54.0%

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

   if 4.9999999999999999e-155 < (*.f64 F F) < 9.99999999999999939e-99

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. associate-*l/ 85.4%

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

      3. *-lft-identity 85.4%

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

      4. sqr-neg 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in l around 0 76.6%

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

      1. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in F around 0 76.7%

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

      1. associate-*r/ 76.7%

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

      2. *-commutative 76.7%

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

      3. neg-mul-1 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. unpow2 76.7%

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

   9. Simplified 76.7%

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

   if 9.99999999999999939e-99 < (*.f64 F F)

   1. Initial program 97.7%

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

      1. sqr-neg 97.7%

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

      2. associate-*l/ 97.7%

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

      3. *-lft-identity 97.7%

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

      4. sqr-neg 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in l around 0 86.8%

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

      1. unpow2 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in F around inf 91.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-98}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5: 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 78.2%

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

   1. sqr-neg 78.2%

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

   2. associate-*l/ 78.2%

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

   3. *-lft-identity 78.2%

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

   4. sqr-neg 78.2%

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

3. Simplified 78.2%

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

4. Taylor expanded in l around 0 67.1%

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

   1. unpow2 67.1%

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

6. Simplified 67.1%

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

7. Taylor expanded in F around inf 75.0%

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

8. Final simplification 75.0%

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

Alternative 6: 3.3% accurate, 311.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (F l) :precision binary64 -1.0)

C

double code(double F, double l) {
	return -1.0;
}

Fortran

real(8) function code(f, l)
    real(8), intent (in) :: f
    real(8), intent (in) :: l
    code = -1.0d0
end function

Java

public static double code(double F, double l) {
	return -1.0;
}

Python

def code(F, l):
	return -1.0

Julia

function code(F, l)
	return -1.0
end

MATLAB

function tmp = code(F, l)
	tmp = -1.0;
end

Wolfram

code[F_, l_] := -1.0

Derivation

1. Initial program 78.2%

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

   1. associate-/r/ 78.2%

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

   2. inv-pow 78.2%

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

   3. pow-to-exp 41.4%

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

   4. associate-/l* 43.0%

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

3. Applied egg-rr 43.0%

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

   1. exp-to-pow 82.2%

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

   2. unpow-1 82.2%

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

   3. clear-num 82.2%

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

   4. associate-/l/ 78.2%

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

   5. add-sqr-sqrt 41.7%

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

   6. frac-times 43.5%

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

   7. div-inv 43.5%

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

   8. associate-*l* 42.4%

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

   9. add-sqr-sqrt 21.6%

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

   10. sqrt-unprod 35.8%

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

   11. frac-times 35.5%

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

   12. add-sqr-sqrt 35.5%

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

   13. associate-/l/ 35.8%

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

   14. clear-num 35.8%

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

5. Applied egg-rr 25.1%

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

   1. *-commutative 25.1%

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

   2. associate-*l/ 25.1%

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

   3. *-lft-identity 25.1%

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

   4. associate-/r/ 25.1%

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

   5. *-inverses 52.1%

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

7. Simplified 52.1%

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

8. Taylor expanded in l around 0 3.3%

   \[\leadsto \color{blue}{-1} \]

9. Final simplification 3.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))