VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 97.5%

Time: 12.1s

Alternatives: 5

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.0412032942225753e-64
  2. l = -536742184210301.5

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: 76.4% 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: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{\pi} \cdot {\pi}^{2}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-16}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -2e+19)
   (* (/ l PI) (pow PI 2.0))
   (if (<= (* PI l) 1e-16) (- (* PI l) (* (/ PI F) (/ l F))) (* PI l))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -2e+19) {
		tmp = (l / ((double) M_PI)) * pow(((double) M_PI), 2.0);
	} else if ((((double) M_PI) * l) <= 1e-16) {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((Math.PI * l) <= -2e+19) {
		tmp = (l / Math.PI) * Math.pow(Math.PI, 2.0);
	} else if ((Math.PI * l) <= 1e-16) {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (math.pi * l) <= -2e+19:
		tmp = (l / math.pi) * math.pow(math.pi, 2.0)
	elif (math.pi * l) <= 1e-16:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= -2e+19)
		tmp = Float64(Float64(l / pi) * (pi ^ 2.0));
	elseif (Float64(pi * l) <= 1e-16)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((pi * l) <= -2e+19)
		tmp = (l / pi) * (pi ^ 2.0);
	elseif ((pi * l) <= 1e-16)
		tmp = (pi * l) - ((pi / F) * (l / F));
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], -2e+19], N[(N[(l / Pi), $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 1e-16], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < -2e19

   1. Initial program 64.9%

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

      1. associate-*l/ 64.9%

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

      2. *-lft-identity 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in l around 0 42.8%

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

      1. unpow2 42.8%

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

   6. Simplified 42.8%

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

      1. add-cbrt-cube 11.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)\right) \cdot \left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}} \]

      2. pow3 11.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right)}^{3}}} \]

   8. Applied egg-rr 14.3%

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

      1. rem-cbrt-cube 46.4%

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

      2. flip-+ 45.3%

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

      3. associate-*r/ 45.4%

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

      4. pow2 45.4%

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

      5. *-commutative 45.4%

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

      6. *-commutative 45.4%

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

      7. swap-sqr 45.4%

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

      8. pow-prod-up 45.4%

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

      9. metadata-eval 45.4%

         \[\leadsto \frac{\ell \cdot \left({\pi}^{2} - {F}^{\color{blue}{-4}} \cdot \left(\pi \cdot \pi\right)\right)}{\pi - \pi \cdot {F}^{-2}} \]

      10. pow2 45.4%

          \[\leadsto \frac{\ell \cdot \left({\pi}^{2} - {F}^{-4} \cdot \color{blue}{{\pi}^{2}}\right)}{\pi - \pi \cdot {F}^{-2}} \]

      11. *-un-lft-identity 45.4%

          \[\leadsto \frac{\ell \cdot \left({\pi}^{2} - {F}^{-4} \cdot {\pi}^{2}\right)}{\color{blue}{1 \cdot \pi} - \pi \cdot {F}^{-2}} \]

      12. *-commutative 45.4%

          \[\leadsto \frac{\ell \cdot \left({\pi}^{2} - {F}^{-4} \cdot {\pi}^{2}\right)}{1 \cdot \pi - \color{blue}{{F}^{-2} \cdot \pi}} \]

   10. Applied egg-rr 45.4%

       \[\leadsto \color{blue}{\frac{\ell \cdot \left({\pi}^{2} - {F}^{-4} \cdot {\pi}^{2}\right)}{\pi \cdot \left(1 - {F}^{-2}\right)}} \]
   11. Step-by-step derivation

       1. times-frac 45.3%

          \[\leadsto \color{blue}{\frac{\ell}{\pi} \cdot \frac{{\pi}^{2} - {F}^{-4} \cdot {\pi}^{2}}{1 - {F}^{-2}}} \]

       2. cancel-sign-sub-inv 45.3%

          \[\leadsto \frac{\ell}{\pi} \cdot \frac{\color{blue}{{\pi}^{2} + \left(-{F}^{-4}\right) \cdot {\pi}^{2}}}{1 - {F}^{-2}} \]

       3. distribute-rgt1-in 45.3%

          \[\leadsto \frac{\ell}{\pi} \cdot \frac{\color{blue}{\left(\left(-{F}^{-4}\right) + 1\right) \cdot {\pi}^{2}}}{1 - {F}^{-2}} \]

   12. Simplified 45.3%

       \[\leadsto \color{blue}{\frac{\ell}{\pi} \cdot \frac{\left(\left(-{F}^{-4}\right) + 1\right) \cdot {\pi}^{2}}{1 - {F}^{-2}}} \]

   13. Taylor expanded in F around inf 99.4%

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

   if -2e19 < (*.f64 (PI.f64) l) < 9.9999999999999998e-17

   1. Initial program 87.4%

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

      1. associate-*l/ 88.0%

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

      2. *-un-lft-identity 88.0%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. times-frac 99.6%

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

   6. Simplified 99.6%

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

   if 9.9999999999999998e-17 < (*.f64 (PI.f64) l)

   1. Initial program 69.6%

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

      1. associate-*l/ 69.6%

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

      2. *-lft-identity 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in l around 0 53.8%

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

      1. unpow2 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in F around inf 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{\pi} \cdot {\pi}^{2}\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-16}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+19) (not (<= (* PI l) 1e-16)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+19) || !((((double) M_PI) * l) <= 1e-16)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+19) || !((Math.PI * l) <= 1e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+19) or not ((math.pi * l) <= 1e-16):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+19) || !(Float64(pi * l) <= 1e-16))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+19) || ~(((pi * l) <= 1e-16)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-16]], $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) < -2e19 or 9.9999999999999998e-17 < (*.f64 (PI.f64) l)

   1. Initial program 67.4%

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

      1. associate-*l/ 67.4%

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

      2. *-lft-identity 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -2e19 < (*.f64 (PI.f64) l) < 9.9999999999999998e-17

   1. Initial program 87.4%

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

      1. associate-*l/ 88.0%

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

      2. *-un-lft-identity 88.0%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. times-frac 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+19) (not (<= (* PI l) 1e-16)))
   (* PI l)
   (* PI (- l (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+19) || !((((double) M_PI) * l) <= 1e-16)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (l - (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+19) || !((Math.PI * l) <= 1e-16)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (l - (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+19) or not ((math.pi * l) <= 1e-16):
		tmp = math.pi * l
	else:
		tmp = math.pi * (l - (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+19) || !(Float64(pi * l) <= 1e-16))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(l - Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+19) || ~(((pi * l) <= 1e-16)))
		tmp = pi * l;
	else
		tmp = pi * (l - (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+19], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-16]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(l - N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e19 or 9.9999999999999998e-17 < (*.f64 (PI.f64) l)

   1. Initial program 67.4%

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

      1. associate-*l/ 67.4%

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

      2. *-lft-identity 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in F around inf 99.5%

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

   if -2e19 < (*.f64 (PI.f64) l) < 9.9999999999999998e-17

   1. Initial program 87.4%

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

      1. associate-*l/ 88.0%

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

      2. *-un-lft-identity 88.0%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. times-frac 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around 0 87.5%

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

      1. distribute-rgt-out-- 87.5%

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

      2. unpow2 87.5%

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

      3. associate-*l/ 88.0%

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

      4. *-commutative 88.0%

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

      5. associate-*l/ 88.0%

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

      6. *-commutative 88.0%

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

      7. distribute-rgt-out-- 88.0%

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

   9. Simplified 88.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+19} \lor \neg \left(\pi \cdot \ell \leq 10^{-16}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \]

Alternative 4: 73.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-103}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (/ PI F) (/ (- l) F))))
   (if (<= (* F F) 5e-283)
     t_0
     (if (<= (* F F) 5e-103)
       (+ (+ (* PI l) 1.0) -1.0)
       (if (<= (* F F) 1e-29) t_0 (* PI l))))))

C

double code(double F, double l) {
	double t_0 = (((double) M_PI) / F) * (-l / F);
	double tmp;
	if ((F * F) <= 5e-283) {
		tmp = t_0;
	} else if ((F * F) <= 5e-103) {
		tmp = ((((double) M_PI) * l) + 1.0) + -1.0;
	} else if ((F * F) <= 1e-29) {
		tmp = t_0;
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = (Math.PI / F) * (-l / F);
	double tmp;
	if ((F * F) <= 5e-283) {
		tmp = t_0;
	} else if ((F * F) <= 5e-103) {
		tmp = ((Math.PI * l) + 1.0) + -1.0;
	} else if ((F * F) <= 1e-29) {
		tmp = t_0;
	} else {
		tmp = Math.PI * l;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = (math.pi / F) * (-l / F)
	tmp = 0
	if (F * F) <= 5e-283:
		tmp = t_0
	elif (F * F) <= 5e-103:
		tmp = ((math.pi * l) + 1.0) + -1.0
	elif (F * F) <= 1e-29:
		tmp = t_0
	else:
		tmp = math.pi * l
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(pi / F) * Float64(Float64(-l) / F))
	tmp = 0.0
	if (Float64(F * F) <= 5e-283)
		tmp = t_0;
	elseif (Float64(F * F) <= 5e-103)
		tmp = Float64(Float64(Float64(pi * l) + 1.0) + -1.0);
	elseif (Float64(F * F) <= 1e-29)
		tmp = t_0;
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = (pi / F) * (-l / F);
	tmp = 0.0;
	if ((F * F) <= 5e-283)
		tmp = t_0;
	elseif ((F * F) <= 5e-103)
		tmp = ((pi * l) + 1.0) + -1.0;
	elseif ((F * F) <= 1e-29)
		tmp = t_0;
	else
		tmp = pi * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(F * F), $MachinePrecision], 5e-283], t$95$0, If[LessEqual[N[(F * F), $MachinePrecision], 5e-103], N[(N[(N[(Pi * l), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-29], t$95$0, N[(Pi * l), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 5.0000000000000001e-283 or 4.99999999999999966e-103 < (*.f64 F F) < 9.99999999999999943e-30

   1. Initial program 46.8%

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

      1. associate-*l/ 47.6%

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

      2. *-lft-identity 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in l around 0 38.6%

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

      1. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in F around 0 39.4%

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

      1. mul-1-neg 39.4%

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

      2. associate-/l* 39.4%

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

      3. unpow2 39.4%

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

      4. associate-/l* 39.4%

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

   9. Simplified 39.4%

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

       1. associate-/r/ 59.3%

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

   11. Applied egg-rr 59.3%

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

   if 5.0000000000000001e-283 < (*.f64 F F) < 4.99999999999999966e-103

   1. Initial program 57.0%

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

      1. associate-*l/ 57.2%

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

      2. *-lft-identity 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in l around 0 32.0%

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

      1. unpow2 32.0%

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

   6. Simplified 32.0%

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

   7. Taylor expanded in F around inf 68.4%

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

      1. *-commutative 68.4%

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

      2. expm1-log1p-u 33.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)} \]

   9. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-udef 33.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \ell\right)} - 1} \]

       2. log1p-udef 33.3%

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

       3. add-exp-log 68.6%

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

   11. Applied egg-rr 68.6%

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

   if 9.99999999999999943e-30 < (*.f64 F F)

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in l around 0 93.7%

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

      1. unpow2 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in F around inf 98.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-103}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \cdot F \leq 10^{-29}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5: 73.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. associate-*l/ 78.5%

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

   2. *-lft-identity 78.5%

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

3. Simplified 78.5%

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

4. Taylor expanded in l around 0 69.5%

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

   1. unpow2 69.5%

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

6. Simplified 69.5%

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

7. Taylor expanded in F around inf 76.9%

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

8. Final simplification 76.9%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))