VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 98.9%

Time: 13.4s

Alternatives: 4

Speedup: 0.7×

• could not determine a ground truth

  1. F = -2.1356821790824245e-204
  2. l = 3178034162763176.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+23) (not (<= (* PI l) 10000000000000.0)))
   (+ (* PI l) -1.0)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+23) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		tmp = (((double) M_PI) * l) + -1.0;
	} 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+23) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = (Math.PI * l) + -1.0;
	} 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+23) or not ((math.pi * l) <= 10000000000000.0):
		tmp = (math.pi * l) + -1.0
	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+23) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(Float64(pi * l) + -1.0);
	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+23) || ~(((pi * l) <= 10000000000000.0)))
		tmp = (pi * l) + -1.0;
	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+23], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + -1.0), $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) < -4.9999999999999999e23 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 62.3%

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

      1. add-cbrt-cube 56.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{\left(\frac{1}{F \cdot F} \cdot \frac{1}{F \cdot F}\right) \cdot \frac{1}{F \cdot F}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. pow3 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{\left(\frac{1}{F \cdot F}\right)}^{3}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      3. inv-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({\left(F \cdot F\right)}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      4. unpow-prod-down 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{-1} \cdot {F}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      5. pow-prod-up 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{\left(-1 + -1\right)}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      6. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\left({F}^{\left(-1 + -1\right)}\right)}^{\color{blue}{\left(2 + 1\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      7. pow-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{F}^{\left(\left(-1 + -1\right) \cdot \left(2 + 1\right)\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      8. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(\color{blue}{-2} \cdot \left(2 + 1\right)\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      9. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(-2 \cdot \color{blue}{3}\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      10. metadata-eval 56.8%

          \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\color{blue}{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Applied egg-rr 56.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{{F}^{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   5. Applied egg-rr 54.9%

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

      1. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   if -4.9999999999999999e23 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 90.6%

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

      1. associate-*l/ 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+23) (not (<= (* PI l) 10000000000000.0)))
   (+ (* PI l) -1.0)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+23) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		tmp = (((double) M_PI) * l) + -1.0;
	} 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+23) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = (Math.PI * l) + -1.0;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+23) or not ((math.pi * l) <= 10000000000000.0):
		tmp = (math.pi * l) + -1.0
	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+23) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(Float64(pi * l) + -1.0);
	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+23) || ~(((pi * l) <= 10000000000000.0)))
		tmp = (pi * l) + -1.0;
	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+23], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + -1.0), $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) < -4.9999999999999999e23 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 62.3%

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

      1. add-cbrt-cube 56.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{\left(\frac{1}{F \cdot F} \cdot \frac{1}{F \cdot F}\right) \cdot \frac{1}{F \cdot F}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. pow3 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{\left(\frac{1}{F \cdot F}\right)}^{3}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      3. inv-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({\left(F \cdot F\right)}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      4. unpow-prod-down 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{-1} \cdot {F}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      5. pow-prod-up 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{\left(-1 + -1\right)}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      6. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\left({F}^{\left(-1 + -1\right)}\right)}^{\color{blue}{\left(2 + 1\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      7. pow-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{F}^{\left(\left(-1 + -1\right) \cdot \left(2 + 1\right)\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      8. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(\color{blue}{-2} \cdot \left(2 + 1\right)\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      9. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(-2 \cdot \color{blue}{3}\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      10. metadata-eval 56.8%

          \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\color{blue}{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Applied egg-rr 56.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{{F}^{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   5. Applied egg-rr 54.9%

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

      1. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   if -4.9999999999999999e23 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 90.6%

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

   2. Taylor expanded in l around 0 88.2%

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

      1. unpow2 88.2%

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

      2. times-frac 96.9%

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

   4. Simplified 96.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+23) (not (<= (* PI l) 10000000000000.0)))
   (+ (* PI l) -1.0)
   (- (* PI l) (/ PI (* F (/ F l))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+23) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		tmp = (((double) M_PI) * l) + -1.0;
	} 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+23) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = (Math.PI * l) + -1.0;
	} else {
		tmp = (Math.PI * l) - (Math.PI / (F * (F / l)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -5e+23) or not ((math.pi * l) <= 10000000000000.0):
		tmp = (math.pi * l) + -1.0
	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+23) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(Float64(pi * l) + -1.0);
	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+23) || ~(((pi * l) <= 10000000000000.0)))
		tmp = (pi * l) + -1.0;
	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+23], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + -1.0), $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) < -4.9999999999999999e23 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 62.3%

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

      1. add-cbrt-cube 56.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{\left(\frac{1}{F \cdot F} \cdot \frac{1}{F \cdot F}\right) \cdot \frac{1}{F \cdot F}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      2. pow3 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{\left(\frac{1}{F \cdot F}\right)}^{3}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      3. inv-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({\left(F \cdot F\right)}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      4. unpow-prod-down 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{-1} \cdot {F}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      5. pow-prod-up 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{\left(-1 + -1\right)}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

      6. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{\left({F}^{\left(-1 + -1\right)}\right)}^{\color{blue}{\left(2 + 1\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      7. pow-pow 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{F}^{\left(\left(-1 + -1\right) \cdot \left(2 + 1\right)\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

      8. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(\color{blue}{-2} \cdot \left(2 + 1\right)\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      9. metadata-eval 56.8%

         \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(-2 \cdot \color{blue}{3}\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

      10. metadata-eval 56.8%

          \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\color{blue}{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Applied egg-rr 56.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{{F}^{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   5. Applied egg-rr 54.9%

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

      1. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   if -4.9999999999999999e23 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 90.6%

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

   2. Taylor expanded in l around 0 88.2%

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

      1. unpow2 88.2%

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

      2. times-frac 96.9%

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

   4. Simplified 96.9%

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

      1. clear-num 96.8%

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

      2. frac-times 97.0%

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

      3. *-un-lft-identity 97.0%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+23} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell + -1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 4: 50.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. add-cbrt-cube 66.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{\left(\frac{1}{F \cdot F} \cdot \frac{1}{F \cdot F}\right) \cdot \frac{1}{F \cdot F}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. pow3 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{\left(\frac{1}{F \cdot F}\right)}^{3}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. inv-pow 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({\left(F \cdot F\right)}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

   4. unpow-prod-down 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{-1} \cdot {F}^{-1}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

   5. pow-prod-up 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{\color{blue}{\left({F}^{\left(-1 + -1\right)}\right)}}^{3}} \cdot \tan \left(\pi \cdot \ell\right) \]

   6. metadata-eval 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{\left({F}^{\left(-1 + -1\right)}\right)}^{\color{blue}{\left(2 + 1\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   7. pow-pow 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{\color{blue}{{F}^{\left(\left(-1 + -1\right) \cdot \left(2 + 1\right)\right)}}} \cdot \tan \left(\pi \cdot \ell\right) \]

   8. metadata-eval 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(\color{blue}{-2} \cdot \left(2 + 1\right)\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

   9. metadata-eval 66.1%

      \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\left(-2 \cdot \color{blue}{3}\right)}} \cdot \tan \left(\pi \cdot \ell\right) \]

   10. metadata-eval 66.1%

       \[\leadsto \pi \cdot \ell - \sqrt[3]{{F}^{\color{blue}{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

3. Applied egg-rr 66.1%

   \[\leadsto \pi \cdot \ell - \color{blue}{\sqrt[3]{{F}^{-6}}} \cdot \tan \left(\pi \cdot \ell\right) \]

5. Applied egg-rr 27.5%

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

   1. *-inverses 49.6%

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

7. Simplified 49.6%

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

8. Final simplification 49.6%

   \[\leadsto \pi \cdot \ell + -1 \]

Reproduce

herbie shell --seed 2023273 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))