VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 81.2%

Time: 22.5s

Alternatives: 5

Speedup: 1.0×

• could not determine a ground truth

  1. F = 3.3241738177838393e-130
  2. l = -1081223097417761.5

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: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\pi \cdot \ell\right)\\ \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot t_0 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{t_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (tan (* PI l))))
   (if (<= (- (* PI l) (* (/ 1.0 (* F F)) t_0)) 5e+287)
     (- (* PI l) (/ (/ t_0 F) F))
     (- (* PI l) (/ (* F F) t_0)))))

C

double code(double F, double l) {
	double t_0 = tan((((double) M_PI) * l));
	double tmp;
	if (((((double) M_PI) * l) - ((1.0 / (F * F)) * t_0)) <= 5e+287) {
		tmp = (((double) M_PI) * l) - ((t_0 / F) / F);
	} else {
		tmp = (((double) M_PI) * l) - ((F * F) / t_0);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.tan((Math.PI * l));
	double tmp;
	if (((Math.PI * l) - ((1.0 / (F * F)) * t_0)) <= 5e+287) {
		tmp = (Math.PI * l) - ((t_0 / F) / F);
	} else {
		tmp = (Math.PI * l) - ((F * F) / t_0);
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = math.tan((math.pi * l))
	tmp = 0
	if ((math.pi * l) - ((1.0 / (F * F)) * t_0)) <= 5e+287:
		tmp = (math.pi * l) - ((t_0 / F) / F)
	else:
		tmp = (math.pi * l) - ((F * F) / t_0)
	return tmp

Julia

function code(F, l)
	t_0 = tan(Float64(pi * l))
	tmp = 0.0
	if (Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * t_0)) <= 5e+287)
		tmp = Float64(Float64(pi * l) - Float64(Float64(t_0 / F) / F));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(F * F) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = tan((pi * l));
	tmp = 0.0;
	if (((pi * l) - ((1.0 / (F * F)) * t_0)) <= 5e+287)
		tmp = (pi * l) - ((t_0 / F) / F);
	else
		tmp = (pi * l) - ((F * F) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 5e+287], N[(N[(Pi * l), $MachinePrecision] - N[(N[(t$95$0 / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(F * F), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 1 (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < 5e287

   1. Initial program 86.8%

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

      1. associate-*l/ 86.9%

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

      2. *-un-lft-identity 86.9%

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

      3. associate-/r* 89.0%

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

   3. Applied egg-rr 89.0%

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

   if 5e287 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 1 (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 28.8%

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

      1. associate-/r/ 28.8%

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

      2. inv-pow 28.8%

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

      3. pow-to-exp 1.6%

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

      4. diff-log 0.0%

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

      5. diff-log 1.6%

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

      6. associate-/l* 1.6%

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

   3. Applied egg-rr 1.6%

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

      1. add-sqr-sqrt 1.6%

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

      2. sqrt-unprod 1.6%

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

      3. *-commutative 1.6%

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

      4. *-commutative 1.6%

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

      5. swap-sqr 1.6%

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

      6. metadata-eval 1.6%

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

      7. *-un-lft-identity 1.6%

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

      8. sqrt-unprod 0.0%

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

   5. Applied egg-rr 59.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{\tan \left(\pi \cdot \ell\right)}\\ \end{array} \]

Alternative 2: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 2e-305)
   (- (* PI l) (* PI (/ (/ l F) F)))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 2e-305) {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * ((l / F) / F));
	} 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 ((F * F) <= 2e-305) {
		tmp = (Math.PI * l) - (Math.PI * ((l / F) / F));
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (F * F) <= 2e-305:
		tmp = (math.pi * l) - (math.pi * ((l / F) / F))
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 2e-305)
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(Float64(l / F) / F)));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 2e-305)
		tmp = (pi * l) - (pi * ((l / F) / F));
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 2e-305], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 1.99999999999999999e-305

   1. Initial program 30.4%

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

   2. Taylor expanded in l around 0 27.8%

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

      1. unpow2 27.8%

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

      2. times-frac 46.8%

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

   4. Simplified 46.8%

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

      1. clear-num 46.8%

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

      2. frac-times 46.8%

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

      3. *-un-lft-identity 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. clear-num 46.7%

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

      2. associate-/r/ 46.7%

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

      3. associate-/r* 46.8%

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

      4. clear-num 46.8%

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

   8. Applied egg-rr 46.8%

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

   if 1.99999999999999999e-305 < (*.f64 F F)

   1. Initial program 93.2%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\pi \cdot \ell\right)\\ \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{t_0}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (tan (* PI l))))
   (if (<= (* F F) 0.0)
     (- (* PI l) (/ (* F F) t_0))
     (- (* PI l) (/ t_0 (* F F))))))

C

double code(double F, double l) {
	double t_0 = tan((((double) M_PI) * l));
	double tmp;
	if ((F * F) <= 0.0) {
		tmp = (((double) M_PI) * l) - ((F * F) / t_0);
	} else {
		tmp = (((double) M_PI) * l) - (t_0 / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.tan((Math.PI * l));
	double tmp;
	if ((F * F) <= 0.0) {
		tmp = (Math.PI * l) - ((F * F) / t_0);
	} else {
		tmp = (Math.PI * l) - (t_0 / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = math.tan((math.pi * l))
	tmp = 0
	if (F * F) <= 0.0:
		tmp = (math.pi * l) - ((F * F) / t_0)
	else:
		tmp = (math.pi * l) - (t_0 / (F * F))
	return tmp

Julia

function code(F, l)
	t_0 = tan(Float64(pi * l))
	tmp = 0.0
	if (Float64(F * F) <= 0.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(F * F) / t_0));
	else
		tmp = Float64(Float64(pi * l) - Float64(t_0 / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = tan((pi * l));
	tmp = 0.0;
	if ((F * F) <= 0.0)
		tmp = (pi * l) - ((F * F) / t_0);
	else
		tmp = (pi * l) - (t_0 / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(F * F), $MachinePrecision], 0.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(F * F), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(t$95$0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 0.0

   1. Initial program 29.0%

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

      1. associate-/r/ 29.0%

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

      2. inv-pow 29.0%

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

      3. pow-to-exp 16.0%

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

      4. diff-log 14.9%

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

      5. diff-log 16.0%

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

      6. associate-/l* 21.7%

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

   3. Applied egg-rr 21.7%

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

      1. add-sqr-sqrt 21.5%

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

      2. sqrt-unprod 21.7%

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

      3. *-commutative 21.7%

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

      4. *-commutative 21.7%

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

      5. swap-sqr 21.7%

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

      6. metadata-eval 21.7%

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

      7. *-un-lft-identity 21.7%

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

      8. sqrt-unprod 0.0%

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

   5. Applied egg-rr 53.8%

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

   if 0.0 < (*.f64 F F)

   1. Initial program 92.3%

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

      1. associate-*l/ 92.4%

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

      2. *-lft-identity 92.4%

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

   3. Simplified 92.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{\tan \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in l around 0 72.1%

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

   1. unpow2 72.1%

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

   2. times-frac 76.7%

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

4. Simplified 76.7%

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

5. Final simplification 76.7%

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

Alternative 5: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in l around 0 72.1%

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

   1. unpow2 72.1%

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

   2. times-frac 76.7%

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

4. Simplified 76.7%

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

   1. clear-num 76.6%

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

   2. frac-times 76.7%

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

   3. *-un-lft-identity 76.7%

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

6. Applied egg-rr 76.7%

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

   1. clear-num 76.6%

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

   2. associate-/r/ 76.6%

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

   3. associate-/r* 76.6%

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

   4. clear-num 76.7%

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

8. Applied egg-rr 76.7%

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

9. Final simplification 76.7%

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

Reproduce

herbie shell --seed 2023252 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))