VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.5%

Time: 14.3s

Alternatives: 8

Speedup: 0.7×

• could not determine a ground truth

  1. F = 6.245059292260807e-107
  2. l = 1973490211324324.5

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 0.0001)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 0.0001)) {
		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+19) || !((Math.PI * l) <= 0.0001)) {
		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+19) or not ((math.pi * l) <= 0.0001):
		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+19) || !(Float64(pi * l) <= 0.0001))
		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+19) || ~(((pi * l) <= 0.0001)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.0001]], $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) < -5e19 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 58.2%

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

      1. sqr-neg 58.2%

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

      2. associate-*l/ 58.2%

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

      3. *-lft-identity 58.2%

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

      4. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in l around 0 45.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5e19 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 91.6%

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

      1. associate-*l/ 92.1%

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

      2. *-un-lft-identity 92.1%

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

      3. associate-/r* 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 4 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-116} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-45}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F \cdot \left(-F\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -4e-37)
   (* PI l)
   (if (<= (* PI l) 4e-298)
     (/ (/ l (/ F PI)) (- F))
     (if (or (<= (* PI l) 5e-116) (not (<= (* PI l) 2e-45)))
       (* PI l)
       (/ (* PI l) (* F (- F)))))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -4e-37) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 4e-298) {
		tmp = (l / (F / ((double) M_PI))) / -F;
	} else if (((((double) M_PI) * l) <= 5e-116) || !((((double) M_PI) * l) <= 2e-45)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) / (F * -F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((Math.PI * l) <= -4e-37) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 4e-298) {
		tmp = (l / (F / Math.PI)) / -F;
	} else if (((Math.PI * l) <= 5e-116) || !((Math.PI * l) <= 2e-45)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) / (F * -F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (math.pi * l) <= -4e-37:
		tmp = math.pi * l
	elif (math.pi * l) <= 4e-298:
		tmp = (l / (F / math.pi)) / -F
	elif ((math.pi * l) <= 5e-116) or not ((math.pi * l) <= 2e-45):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) / (F * -F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= -4e-37)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 4e-298)
		tmp = Float64(Float64(l / Float64(F / pi)) / Float64(-F));
	elseif ((Float64(pi * l) <= 5e-116) || !(Float64(pi * l) <= 2e-45))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) / Float64(F * Float64(-F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((pi * l) <= -4e-37)
		tmp = pi * l;
	elseif ((pi * l) <= 4e-298)
		tmp = (l / (F / pi)) / -F;
	elseif (((pi * l) <= 5e-116) || ~(((pi * l) <= 2e-45)))
		tmp = pi * l;
	else
		tmp = (pi * l) / (F * -F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], -4e-37], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 4e-298], N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision], If[Or[LessEqual[N[(Pi * l), $MachinePrecision], 5e-116], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-45]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] / N[(F * (-F)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < -4.00000000000000027e-37 or 3.99999999999999965e-298 < (*.f64 (PI.f64) l) < 5.0000000000000003e-116 or 1.99999999999999997e-45 < (*.f64 (PI.f64) l)

   1. Initial program 68.9%

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

      1. sqr-neg 68.9%

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

      2. associate-*l/ 69.2%

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

      3. *-lft-identity 69.2%

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

      4. sqr-neg 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in l around 0 60.6%

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

   5. Taylor expanded in F around inf 91.6%

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

   if -4.00000000000000027e-37 < (*.f64 (PI.f64) l) < 3.99999999999999965e-298

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

      4. sqr-neg 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 90.7%

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

   5. Taylor expanded in F around 0 48.9%

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

      1. associate-*r/ 48.9%

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

      2. *-commutative 48.9%

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

      3. neg-mul-1 48.9%

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

      4. *-commutative 48.9%

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

      5. distribute-rgt-neg-in 48.9%

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

   7. Simplified 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/l* 48.8%

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

      3. unpow2 48.8%

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

      4. associate-*l/ 57.9%

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

      5. distribute-neg-frac 57.9%

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

      6. associate-/r* 57.9%

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

      7. un-div-inv 57.9%

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

      8. *-commutative 57.9%

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

      9. distribute-lft-neg-in 57.9%

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

      10. distribute-neg-frac 57.9%

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

      11. metadata-eval 57.9%

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

      12. div-inv 57.9%

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

      13. clear-num 57.9%

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

   9. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. frac-2neg 57.9%

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

       3. metadata-eval 57.9%

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

       4. un-div-inv 57.9%

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

       5. associate-*r/ 58.0%

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

       6. *-commutative 58.0%

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

       7. associate-/l* 57.9%

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

   11. Applied egg-rr 57.9%

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

   if 5.0000000000000003e-116 < (*.f64 (PI.f64) l) < 1.99999999999999997e-45

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in l around 0 99.9%

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

   5. Taylor expanded in F around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.1%

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

      3. unpow2 99.1%

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

      4. associate-*l/ 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. associate-/r* 99.3%

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

      7. un-div-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. distribute-neg-frac 99.7%

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

      11. metadata-eval 99.7%

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

      12. div-inv 99.6%

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

      13. clear-num 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. clear-num 99.7%

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

       2. associate-*r/ 99.6%

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

       3. frac-times 99.9%

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

       4. *-un-lft-identity 99.9%

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

       5. div-inv 99.9%

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

       6. metadata-eval 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 4 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-116} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-45}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F \cdot \left(-F\right)}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, -\frac{\frac{\pi \cdot \ell}{F}}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 0.0001)))
   (* PI l)
   (fma PI l (- (/ (/ (* PI l) F) F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 0.0001)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = fma(((double) M_PI), l, -(((((double) M_PI) * l) / F) / F));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+19) || !(Float64(pi * l) <= 0.0001))
		tmp = Float64(pi * l);
	else
		tmp = fma(pi, l, Float64(-Float64(Float64(Float64(pi * l) / F) / F)));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * l + (-N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e19 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 58.2%

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

      1. sqr-neg 58.2%

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

      2. associate-*l/ 58.2%

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

      3. *-lft-identity 58.2%

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

      4. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in l around 0 45.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5e19 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

      2. sqr-neg 91.6%

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

      3. *-commutative 91.6%

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

      4. fma-neg 91.6%

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

      5. associate-*l/ 92.1%

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

      6. times-frac 99.7%

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

      7. distribute-lft-neg-in 99.7%

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

      8. neg-mul-1 99.7%

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

      9. associate-/r* 99.7%

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

      10. metadata-eval 99.7%

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

      11. distribute-neg-frac 99.7%

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

      12. metadata-eval 99.7%

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

      13. times-frac 92.1%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in l around 0 99.4%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-1 \cdot \frac{\ell \cdot \pi}{F}}}{F}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 99.4%

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

      2. *-commutative 99.4%

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

      3. neg-mul-1 99.4%

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

      4. *-commutative 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, -\frac{\frac{\pi \cdot \ell}{F}}{F}\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 0.0001)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 0.0001)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -5e+19) || !((Math.PI * l) <= 0.0001)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+19) or not ((math.pi * l) <= 0.0001):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+19) || !(Float64(pi * l) <= 0.0001))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+19) || ~(((pi * l) <= 0.0001)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e19 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 58.2%

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

      1. sqr-neg 58.2%

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

      2. associate-*l/ 58.2%

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

      3. *-lft-identity 58.2%

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

      4. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in l around 0 45.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5e19 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 91.6%

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

      1. sqr-neg 91.6%

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

      2. associate-*l/ 92.1%

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

      3. *-lft-identity 92.1%

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

      4. sqr-neg 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in l around 0 91.8%

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

      1. *-commutative 91.8%

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

      2. times-frac 99.4%

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

   6. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 5: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+19) (not (<= (* PI l) 0.0001)))
   (* PI l)
   (- (* PI l) (/ (/ l (/ F PI)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+19) || !((((double) M_PI) * l) <= 0.0001)) {
		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+19) || !((Math.PI * l) <= 0.0001)) {
		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+19) or not ((math.pi * l) <= 0.0001):
		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+19) || !(Float64(pi * l) <= 0.0001))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / Float64(F / pi)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -5e+19) || ~(((pi * l) <= 0.0001)))
		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+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e19 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 58.2%

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

      1. sqr-neg 58.2%

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

      2. associate-*l/ 58.2%

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

      3. *-lft-identity 58.2%

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

      4. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in l around 0 45.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -5e19 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 91.6%

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

      1. associate-*l/ 92.1%

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

      2. *-un-lft-identity 92.1%

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

      3. associate-/r* 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in l around 0 99.4%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 6: 74.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-38}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-115} \lor \neg \left(\ell \leq 6.2 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{F} \cdot \left(\pi \cdot \frac{\ell}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -6e-38)
   (* PI l)
   (if (<= l 5.7e-297)
     (/ (/ l (/ F PI)) (- F))
     (if (or (<= l 7.2e-115) (not (<= l 6.2e-46)))
       (* PI l)
       (* (/ -1.0 F) (* PI (/ l F)))))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -6e-38) {
		tmp = ((double) M_PI) * l;
	} else if (l <= 5.7e-297) {
		tmp = (l / (F / ((double) M_PI))) / -F;
	} else if ((l <= 7.2e-115) || !(l <= 6.2e-46)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (-1.0 / F) * (((double) M_PI) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -6e-38) {
		tmp = Math.PI * l;
	} else if (l <= 5.7e-297) {
		tmp = (l / (F / Math.PI)) / -F;
	} else if ((l <= 7.2e-115) || !(l <= 6.2e-46)) {
		tmp = Math.PI * l;
	} else {
		tmp = (-1.0 / F) * (Math.PI * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -6e-38:
		tmp = math.pi * l
	elif l <= 5.7e-297:
		tmp = (l / (F / math.pi)) / -F
	elif (l <= 7.2e-115) or not (l <= 6.2e-46):
		tmp = math.pi * l
	else:
		tmp = (-1.0 / F) * (math.pi * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -6e-38)
		tmp = Float64(pi * l);
	elseif (l <= 5.7e-297)
		tmp = Float64(Float64(l / Float64(F / pi)) / Float64(-F));
	elseif ((l <= 7.2e-115) || !(l <= 6.2e-46))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(-1.0 / F) * Float64(pi * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -6e-38)
		tmp = pi * l;
	elseif (l <= 5.7e-297)
		tmp = (l / (F / pi)) / -F;
	elseif ((l <= 7.2e-115) || ~((l <= 6.2e-46)))
		tmp = pi * l;
	else
		tmp = (-1.0 / F) * (pi * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -6e-38], N[(Pi * l), $MachinePrecision], If[LessEqual[l, 5.7e-297], N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision], If[Or[LessEqual[l, 7.2e-115], N[Not[LessEqual[l, 6.2e-46]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(-1.0 / F), $MachinePrecision] * N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.99999999999999977e-38 or 5.6999999999999997e-297 < l < 7.20000000000000018e-115 or 6.2000000000000002e-46 < l

   1. Initial program 68.9%

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

      1. sqr-neg 68.9%

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

      2. associate-*l/ 69.2%

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

      3. *-lft-identity 69.2%

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

      4. sqr-neg 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in l around 0 60.6%

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

   5. Taylor expanded in F around inf 91.6%

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

   if -5.99999999999999977e-38 < l < 5.6999999999999997e-297

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

      4. sqr-neg 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in l around 0 90.7%

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

   5. Taylor expanded in F around 0 48.9%

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

      1. associate-*r/ 48.9%

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

      2. *-commutative 48.9%

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

      3. neg-mul-1 48.9%

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

      4. *-commutative 48.9%

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

      5. distribute-rgt-neg-in 48.9%

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

   7. Simplified 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/l* 48.8%

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

      3. unpow2 48.8%

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

      4. associate-*l/ 57.9%

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

      5. distribute-neg-frac 57.9%

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

      6. associate-/r* 57.9%

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

      7. un-div-inv 57.9%

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

      8. *-commutative 57.9%

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

      9. distribute-lft-neg-in 57.9%

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

      10. distribute-neg-frac 57.9%

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

      11. metadata-eval 57.9%

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

      12. div-inv 57.9%

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

      13. clear-num 57.9%

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

   9. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. frac-2neg 57.9%

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

       3. metadata-eval 57.9%

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

       4. un-div-inv 57.9%

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

       5. associate-*r/ 58.0%

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

       6. *-commutative 58.0%

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

       7. associate-/l* 57.9%

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

   11. Applied egg-rr 57.9%

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

   if 7.20000000000000018e-115 < l < 6.2000000000000002e-46

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in l around 0 99.9%

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

   5. Taylor expanded in F around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.1%

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

      3. unpow2 99.1%

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

      4. associate-*l/ 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. associate-/r* 99.3%

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

      7. un-div-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. distribute-neg-frac 99.7%

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

      11. metadata-eval 99.7%

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

      12. div-inv 99.6%

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

      13. clear-num 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-38}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-115} \lor \neg \left(\ell \leq 6.2 \cdot 10^{-46}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{F} \cdot \left(\pi \cdot \frac{\ell}{F}\right)\\ \end{array} \]

Alternative 7: 74.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-37} \lor \neg \left(\ell \leq 4.5 \cdot 10^{-296} \lor \neg \left(\ell \leq 1.2 \cdot 10^{-114}\right) \land \ell \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.1e-37)
         (not (or (<= l 4.5e-296) (and (not (<= l 1.2e-114)) (<= l 2.1e-45)))))
   (* PI l)
   (/ (/ l (/ F PI)) (- F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.1e-37) || !((l <= 4.5e-296) || (!(l <= 1.2e-114) && (l <= 2.1e-45)))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l / (F / ((double) M_PI))) / -F;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1.1e-37) || !((l <= 4.5e-296) || (!(l <= 1.2e-114) && (l <= 2.1e-45)))) {
		tmp = Math.PI * l;
	} else {
		tmp = (l / (F / Math.PI)) / -F;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1.1e-37) or not ((l <= 4.5e-296) or (not (l <= 1.2e-114) and (l <= 2.1e-45))):
		tmp = math.pi * l
	else:
		tmp = (l / (F / math.pi)) / -F
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1.1e-37) || !((l <= 4.5e-296) || (!(l <= 1.2e-114) && (l <= 2.1e-45))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l / Float64(F / pi)) / Float64(-F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1.1e-37) || ~(((l <= 4.5e-296) || (~((l <= 1.2e-114)) && (l <= 2.1e-45)))))
		tmp = pi * l;
	else
		tmp = (l / (F / pi)) / -F;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.1e-37], N[Not[Or[LessEqual[l, 4.5e-296], And[N[Not[LessEqual[l, 1.2e-114]], $MachinePrecision], LessEqual[l, 2.1e-45]]]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.10000000000000001e-37 or 4.5000000000000002e-296 < l < 1.2000000000000001e-114 or 2.09999999999999995e-45 < l

   1. Initial program 68.9%

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

      1. sqr-neg 68.9%

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

      2. associate-*l/ 69.2%

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

      3. *-lft-identity 69.2%

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

      4. sqr-neg 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in l around 0 60.6%

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

   5. Taylor expanded in F around inf 91.6%

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

   if -1.10000000000000001e-37 < l < 4.5000000000000002e-296 or 1.2000000000000001e-114 < l < 2.09999999999999995e-45

   1. Initial program 92.2%

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

      1. sqr-neg 92.2%

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

      2. associate-*l/ 92.3%

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

      3. *-lft-identity 92.3%

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

      4. sqr-neg 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in l around 0 92.3%

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

   5. Taylor expanded in F around 0 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-commutative 58.1%

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

      3. neg-mul-1 58.1%

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

      4. *-commutative 58.1%

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

      5. distribute-rgt-neg-in 58.1%

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

   7. Simplified 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-/l* 57.8%

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

      3. unpow2 57.8%

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

      4. associate-*l/ 65.4%

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

      5. distribute-neg-frac 65.4%

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

      6. associate-/r* 65.4%

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

      7. un-div-inv 65.4%

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

      8. *-commutative 65.4%

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

      9. distribute-lft-neg-in 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

      12. div-inv 65.4%

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

      13. clear-num 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. *-commutative 65.4%

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

       2. frac-2neg 65.4%

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

       3. metadata-eval 65.4%

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

       4. un-div-inv 65.4%

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

       5. associate-*r/ 65.5%

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

       6. *-commutative 65.5%

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

       7. associate-/l* 65.4%

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

   11. Applied egg-rr 65.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-37} \lor \neg \left(\ell \leq 4.5 \cdot 10^{-296} \lor \neg \left(\ell \leq 1.2 \cdot 10^{-114}\right) \land \ell \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{F}{\pi}}}{-F}\\ \end{array} \]

Alternative 8: 73.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. sqr-neg 74.5%

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

   2. associate-*l/ 74.7%

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

   3. *-lft-identity 74.7%

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

   4. sqr-neg 74.7%

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

3. Simplified 74.7%

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

4. Taylor expanded in l around 0 68.2%

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

5. Taylor expanded in F around inf 78.7%

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

6. Final simplification 78.7%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))