VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.2% → 99.3%

Time: 15.7s

Alternatives: 9

Speedup: 0.8×

• could not determine a ground truth

  1. F = 1.2434016974037867e+101
  2. l = -2050487923046418.5

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+18) (not (<= (* PI l) 1e+14)))
   (* PI l)
   (+ (* PI l) (* (/ (tan (* PI l)) F) (/ -1.0 F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+18) || !((((double) M_PI) * l) <= 1e+14)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) / F) * (-1.0 / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+18) || !((Math.PI * l) <= 1e+14)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) / F) * (-1.0 / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+18) or not ((math.pi * l) <= 1e+14):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) / F) * (-1.0 / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+18) || !(Float64(pi * l) <= 1e+14))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) / F) * Float64(-1.0 / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+18) || ~(((pi * l) <= 1e+14)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) / F) * (-1.0 / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e+14]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e18 or 1e14 < (*.f64 (PI.f64) l)

   1. Initial program 60.9%

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

   2. Taylor expanded in F around 0 60.9%

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

      1. unpow-1 60.9%

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

      2. exp-to-pow 33.5%

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

      3. exp-prod 33.5%

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

      4. associate-*l* 33.5%

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

      5. metadata-eval 33.5%

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

      6. exp-to-pow 60.9%

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

   4. Simplified 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-flip 60.9%

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

      3. pow2 60.9%

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

      4. associate-/r/ 60.9%

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

      5. inv-pow 60.9%

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

      6. associate-/l* 60.9%

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

      7. exp-to-pow 32.3%

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

      8. add-sqr-sqrt 4.8%

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

      9. sqrt-unprod 14.9%

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

      10. *-commutative 14.9%

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

      11. *-commutative 14.9%

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

      12. swap-sqr 14.9%

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

   6. Applied egg-rr 47.9%

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

      1. *-inverses 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in l around inf 99.7%

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

   if -5e18 < (*.f64 (PI.f64) l) < 1e14

   1. Initial program 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

   3. Simplified 86.2%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+18) (not (<= (* PI l) 1e+14)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+18) || !((((double) M_PI) * l) <= 1e+14)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+18) || !((Math.PI * l) <= 1e+14)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+18) or not ((math.pi * l) <= 1e+14):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+18) || !(Float64(pi * l) <= 1e+14))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+18) || ~(((pi * l) <= 1e+14)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e+14]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e18 or 1e14 < (*.f64 (PI.f64) l)

   1. Initial program 60.9%

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

   2. Taylor expanded in F around 0 60.9%

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

      1. unpow-1 60.9%

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

      2. exp-to-pow 33.5%

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

      3. exp-prod 33.5%

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

      4. associate-*l* 33.5%

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

      5. metadata-eval 33.5%

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

      6. exp-to-pow 60.9%

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

   4. Simplified 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-flip 60.9%

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

      3. pow2 60.9%

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

      4. associate-/r/ 60.9%

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

      5. inv-pow 60.9%

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

      6. associate-/l* 60.9%

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

      7. exp-to-pow 32.3%

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

      8. add-sqr-sqrt 4.8%

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

      9. sqrt-unprod 14.9%

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

      10. *-commutative 14.9%

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

      11. *-commutative 14.9%

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

      12. swap-sqr 14.9%

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

   6. Applied egg-rr 47.9%

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

      1. *-inverses 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in l around inf 99.7%

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

   if -5e18 < (*.f64 (PI.f64) l) < 1e14

   1. Initial program 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-un-lft-identity 86.2%

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

      3. associate-/r* 98.4%

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

   3. Applied egg-rr 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+18) (not (<= (* PI l) 0.01)))
   (* PI l)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+18) || !((((double) M_PI) * l) <= 0.01)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / F) * (((double) M_PI) / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+18) || !((Math.PI * l) <= 0.01)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+18) or not ((math.pi * l) <= 0.01):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / F) * (math.pi / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+18) || !(Float64(pi * l) <= 0.01))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / F) * Float64(pi / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+18) || ~(((pi * l) <= 0.01)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) * (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e18 or 0.0100000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   2. Taylor expanded in F around 0 61.5%

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

      1. unpow-1 61.5%

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

      2. exp-to-pow 34.0%

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

      3. exp-prod 34.0%

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

      4. associate-*l* 34.0%

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

      5. metadata-eval 34.0%

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

      6. exp-to-pow 61.5%

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

   4. Simplified 61.5%

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

      1. metadata-eval 61.5%

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

      2. pow-flip 61.5%

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

      3. pow2 61.5%

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

      4. associate-/r/ 61.5%

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

      5. inv-pow 61.5%

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

      6. associate-/l* 61.5%

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

      7. exp-to-pow 32.1%

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

      8. add-sqr-sqrt 4.7%

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

      9. sqrt-unprod 14.5%

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

      10. *-commutative 14.5%

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

      11. *-commutative 14.5%

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

      12. swap-sqr 14.5%

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

   6. Applied egg-rr 46.7%

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

      1. *-inverses 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in l around inf 98.9%

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

   if -5e18 < (*.f64 (PI.f64) l) < 0.0100000000000000002

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0 84.7%

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

      1. unpow2 84.7%

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

      2. times-frac 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 4: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+18) (not (<= (* PI l) 0.01)))
   (* PI l)
   (- (* PI l) (/ PI (* F (/ F l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+18) || !((((double) M_PI) * l) <= 0.01)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F * (F / l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+18) || !((Math.PI * l) <= 0.01)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI / (F * (F / l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+18) or not ((math.pi * l) <= 0.01):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi / (F * (F / l)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+18) || !(Float64(pi * l) <= 0.01))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F * Float64(F / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+18) || ~(((pi * l) <= 0.01)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi / (F * (F / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e18 or 0.0100000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   2. Taylor expanded in F around 0 61.5%

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

      1. unpow-1 61.5%

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

      2. exp-to-pow 34.0%

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

      3. exp-prod 34.0%

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

      4. associate-*l* 34.0%

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

      5. metadata-eval 34.0%

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

      6. exp-to-pow 61.5%

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

   4. Simplified 61.5%

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

      1. metadata-eval 61.5%

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

      2. pow-flip 61.5%

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

      3. pow2 61.5%

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

      4. associate-/r/ 61.5%

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

      5. inv-pow 61.5%

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

      6. associate-/l* 61.5%

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

      7. exp-to-pow 32.1%

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

      8. add-sqr-sqrt 4.7%

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

      9. sqrt-unprod 14.5%

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

      10. *-commutative 14.5%

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

      11. *-commutative 14.5%

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

      12. swap-sqr 14.5%

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

   6. Applied egg-rr 46.7%

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

      1. *-inverses 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in l around inf 98.9%

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

   if -5e18 < (*.f64 (PI.f64) l) < 0.0100000000000000002

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0 84.7%

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

      1. unpow2 84.7%

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

      2. times-frac 97.3%

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

   4. Simplified 97.3%

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

      1. clear-num 97.2%

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

      2. frac-times 97.3%

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

      3. *-un-lft-identity 97.3%

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

   6. Applied egg-rr 97.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 5: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+18) (not (<= (* PI l) 0.01)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+18) || !((((double) M_PI) * l) <= 0.01)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+18) || !((Math.PI * l) <= 0.01)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+18) or not ((math.pi * l) <= 0.01):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+18) || !(Float64(pi * l) <= 0.01))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+18) || ~(((pi * l) <= 0.01)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e18 or 0.0100000000000000002 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   2. Taylor expanded in F around 0 61.5%

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

      1. unpow-1 61.5%

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

      2. exp-to-pow 34.0%

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

      3. exp-prod 34.0%

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

      4. associate-*l* 34.0%

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

      5. metadata-eval 34.0%

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

      6. exp-to-pow 61.5%

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

   4. Simplified 61.5%

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

      1. metadata-eval 61.5%

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

      2. pow-flip 61.5%

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

      3. pow2 61.5%

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

      4. associate-/r/ 61.5%

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

      5. inv-pow 61.5%

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

      6. associate-/l* 61.5%

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

      7. exp-to-pow 32.1%

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

      8. add-sqr-sqrt 4.7%

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

      9. sqrt-unprod 14.5%

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

      10. *-commutative 14.5%

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

      11. *-commutative 14.5%

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

      12. swap-sqr 14.5%

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

   6. Applied egg-rr 46.7%

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

      1. *-inverses 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in l around inf 98.9%

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

   if -5e18 < (*.f64 (PI.f64) l) < 0.0100000000000000002

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0 84.7%

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

      1. unpow2 84.7%

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

      2. times-frac 97.3%

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

   4. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. clear-num 97.2%

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

      3. un-div-inv 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in l around 0 84.7%

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

      1. unpow2 84.7%

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

   9. Simplified 84.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.01\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 6: 74.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 1.7 \cdot 10^{-184}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 6.4 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 1.7e-184)
   (+ (* PI l) -1.0)
   (if (<= (* F F) 6.4e-76) (- (/ l (/ (* F F) PI))) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 1.7e-184) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if ((F * F) <= 6.4e-76) {
		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) <= 1.7e-184) {
		tmp = (Math.PI * l) + -1.0;
	} else if ((F * F) <= 6.4e-76) {
		tmp = -(l / ((F * F) / Math.PI));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 1.7e-184:
		tmp = (math.pi * l) + -1.0
	elif (F * F) <= 6.4e-76:
		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) <= 1.7e-184)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (Float64(F * F) <= 6.4e-76)
		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) <= 1.7e-184)
		tmp = (pi * l) + -1.0;
	elseif ((F * F) <= 6.4e-76)
		tmp = -(l / ((F * F) / pi));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 1.7e-184], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 6.4e-76], (-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) < 1.70000000000000002e-184

   1. Initial program 34.1%

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

   2. Taylor expanded in F around 0 34.1%

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

      1. unpow-1 34.1%

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

      2. exp-to-pow 22.0%

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

      3. exp-prod 22.0%

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

      4. associate-*l* 22.0%

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

      5. metadata-eval 22.0%

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

      6. exp-to-pow 34.1%

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

   4. Simplified 34.1%

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

      1. metadata-eval 34.1%

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

      2. pow-flip 34.1%

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

      3. pow2 34.1%

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

      4. associate-/r/ 34.1%

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

      5. inv-pow 34.1%

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

      6. associate-/l* 50.7%

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

      7. exp-to-pow 24.1%

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

      8. add-sqr-sqrt 22.9%

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

      9. sqrt-unprod 23.2%

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

      10. *-commutative 23.2%

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

      11. *-commutative 23.2%

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

      12. swap-sqr 23.2%

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

   6. Applied egg-rr 24.1%

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

      1. *-inverses 52.6%

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

   8. Simplified 52.6%

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

   if 1.70000000000000002e-184 < (*.f64 F F) < 6.3999999999999995e-76

   1. Initial program 79.9%

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

   2. Taylor expanded in l around 0 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 66.3%

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

   4. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.3%

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

      3. un-div-inv 66.3%

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

   6. Applied egg-rr 66.3%

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

   7. Taylor expanded in F around 0 66.3%

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

      1. mul-1-neg 66.3%

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

      2. unpow2 66.3%

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

      3. associate-/l* 66.2%

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

   9. Simplified 66.2%

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

   if 6.3999999999999995e-76 < (*.f64 F F)

   1. Initial program 98.7%

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

   2. Taylor expanded in F around 0 98.7%

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

      1. unpow-1 98.7%

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

      2. exp-to-pow 48.1%

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

      3. exp-prod 48.1%

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

      4. associate-*l* 48.1%

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

      5. metadata-eval 48.1%

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

      6. exp-to-pow 98.7%

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

   4. Simplified 98.7%

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

      1. metadata-eval 98.7%

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

      2. pow-flip 98.7%

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

      3. pow2 98.7%

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

      4. associate-/r/ 98.7%

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

      5. inv-pow 98.7%

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

      6. associate-/l* 98.7%

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

      7. exp-to-pow 49.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. sqrt-unprod 10.4%

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

      10. *-commutative 10.4%

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

      11. *-commutative 10.4%

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

      12. swap-sqr 10.4%

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

   6. Applied egg-rr 26.2%

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

      1. *-inverses 53.3%

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

   8. Simplified 53.3%

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

   9. Taylor expanded in l around inf 97.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 1.7 \cdot 10^{-184}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 6.4 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-190)
   (+ (* PI l) -1.0)
   (if (<= (* F F) 2e-80) (* (/ l F) (/ (- PI) F)) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-190) {
		tmp = (((double) M_PI) * l) + -1.0;
	} else if ((F * F) <= 2e-80) {
		tmp = (l / F) * (-((double) M_PI) / F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-190) {
		tmp = (Math.PI * l) + -1.0;
	} else if ((F * F) <= 2e-80) {
		tmp = (l / F) * (-Math.PI / F);
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-190:
		tmp = (math.pi * l) + -1.0
	elif (F * F) <= 2e-80:
		tmp = (l / F) * (-math.pi / F)
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-190)
		tmp = Float64(Float64(pi * l) + -1.0);
	elseif (Float64(F * F) <= 2e-80)
		tmp = Float64(Float64(l / F) * Float64(Float64(-pi) / F));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-190)
		tmp = (pi * l) + -1.0;
	elseif ((F * F) <= 2e-80)
		tmp = (l / F) * (-pi / F);
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-190], N[(N[(Pi * l), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 2e-80], N[(N[(l / F), $MachinePrecision] * N[((-Pi) / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 2e-190

   1. Initial program 34.1%

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

   2. Taylor expanded in F around 0 34.1%

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

      1. unpow-1 34.1%

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

      2. exp-to-pow 22.0%

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

      3. exp-prod 22.0%

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

      4. associate-*l* 22.0%

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

      5. metadata-eval 22.0%

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

      6. exp-to-pow 34.1%

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

   4. Simplified 34.1%

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

      1. metadata-eval 34.1%

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

      2. pow-flip 34.1%

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

      3. pow2 34.1%

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

      4. associate-/r/ 34.1%

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

      5. inv-pow 34.1%

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

      6. associate-/l* 50.7%

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

      7. exp-to-pow 24.1%

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

      8. add-sqr-sqrt 22.9%

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

      9. sqrt-unprod 23.2%

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

      10. *-commutative 23.2%

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

      11. *-commutative 23.2%

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

      12. swap-sqr 23.2%

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

   6. Applied egg-rr 24.1%

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

      1. *-inverses 52.6%

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

   8. Simplified 52.6%

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

   if 2e-190 < (*.f64 F F) < 1.99999999999999992e-80

   1. Initial program 79.9%

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

   2. Taylor expanded in l around 0 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 66.3%

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

   4. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.3%

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

      3. un-div-inv 66.3%

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

   6. Applied egg-rr 66.3%

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

   7. Taylor expanded in F around 0 66.3%

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

      1. mul-1-neg 66.3%

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

      2. unpow2 66.3%

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

      3. times-frac 66.3%

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

      4. distribute-rgt-neg-in 66.3%

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

      5. distribute-frac-neg 66.3%

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

   9. Simplified 66.3%

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

   if 1.99999999999999992e-80 < (*.f64 F F)

   1. Initial program 98.7%

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

   2. Taylor expanded in F around 0 98.7%

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

      1. unpow-1 98.7%

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

      2. exp-to-pow 48.1%

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

      3. exp-prod 48.1%

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

      4. associate-*l* 48.1%

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

      5. metadata-eval 48.1%

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

      6. exp-to-pow 98.7%

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

   4. Simplified 98.7%

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

      1. metadata-eval 98.7%

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

      2. pow-flip 98.7%

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

      3. pow2 98.7%

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

      4. associate-/r/ 98.7%

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

      5. inv-pow 98.7%

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

      6. associate-/l* 98.7%

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

      7. exp-to-pow 49.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. sqrt-unprod 10.4%

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

      10. *-commutative 10.4%

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

      11. *-commutative 10.4%

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

      12. swap-sqr 10.4%

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

   6. Applied egg-rr 26.2%

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

      1. *-inverses 53.3%

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

   8. Simplified 53.3%

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

   9. Taylor expanded in l around inf 97.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8: 73.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in F around 0 73.6%

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

   1. unpow-1 73.6%

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

   2. exp-to-pow 37.8%

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

   3. exp-prod 37.8%

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

   4. associate-*l* 37.8%

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

   5. metadata-eval 37.8%

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

   6. exp-to-pow 73.6%

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

4. Simplified 73.6%

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

   1. metadata-eval 73.6%

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

   2. pow-flip 73.6%

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

   3. pow2 73.6%

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

   4. associate-/r/ 73.7%

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

   5. inv-pow 73.7%

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

   6. associate-/l* 79.8%

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

   7. exp-to-pow 39.5%

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

   8. add-sqr-sqrt 10.8%

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

   9. sqrt-unprod 16.1%

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

   10. *-commutative 16.1%

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

   11. *-commutative 16.1%

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

   12. swap-sqr 16.1%

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

6. Applied egg-rr 24.8%

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

   1. *-inverses 52.1%

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

8. Simplified 52.1%

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

9. Taylor expanded in l around inf 77.1%

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

10. Final simplification 77.1%

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

Alternative 9: 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 73.6%

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

2. Taylor expanded in F around 0 73.6%

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

   1. unpow-1 73.6%

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

   2. exp-to-pow 37.8%

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

   3. exp-prod 37.8%

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

   4. associate-*l* 37.8%

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

   5. metadata-eval 37.8%

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

   6. exp-to-pow 73.6%

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

4. Simplified 73.6%

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

   1. metadata-eval 73.6%

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

   2. pow-flip 73.6%

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

   3. pow2 73.6%

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

   4. associate-/r/ 73.7%

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

   5. inv-pow 73.7%

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

   6. associate-/l* 79.8%

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

   7. exp-to-pow 39.5%

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

   8. add-sqr-sqrt 10.8%

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

   9. sqrt-unprod 16.1%

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

   10. *-commutative 16.1%

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

   11. *-commutative 16.1%

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

   12. swap-sqr 16.1%

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

6. Applied egg-rr 24.8%

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

   1. *-inverses 52.1%

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

8. Simplified 52.1%

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

9. Taylor expanded in l around 0 3.3%

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

10. Final simplification 3.3%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))