VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 80.6%

Time: 24.3s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = 4.395260740562265e-135
  2. l = -3336118793151274.5

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: 76.3% 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: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} t_0 := \tan \left(\pi \cdot \ell\right)\\ \mathbf{if}\;F \leq 2.25 \cdot 10^{-197}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\pi \cdot \frac{\ell}{F}}}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{t_0}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (let* ((t_0 (tan (* PI l))))
   (if (<= F 2.25e-197)
     (+ (* PI l) (/ -1.0 (/ F (* PI (/ l F)))))
     (if (<= F 5e-149)
       (- (* PI l) (/ (* F F) t_0))
       (- (* PI l) (/ t_0 (* F F)))))))

C

F = abs(F);
double code(double F, double l) {
	double t_0 = tan((((double) M_PI) * l));
	double tmp;
	if (F <= 2.25e-197) {
		tmp = (((double) M_PI) * l) + (-1.0 / (F / (((double) M_PI) * (l / F))));
	} else if (F <= 5e-149) {
		tmp = (((double) M_PI) * l) - ((F * F) / t_0);
	} else {
		tmp = (((double) M_PI) * l) - (t_0 / (F * F));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double t_0 = Math.tan((Math.PI * l));
	double tmp;
	if (F <= 2.25e-197) {
		tmp = (Math.PI * l) + (-1.0 / (F / (Math.PI * (l / F))));
	} else if (F <= 5e-149) {
		tmp = (Math.PI * l) - ((F * F) / t_0);
	} else {
		tmp = (Math.PI * l) - (t_0 / (F * F));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	t_0 = math.tan((math.pi * l))
	tmp = 0
	if F <= 2.25e-197:
		tmp = (math.pi * l) + (-1.0 / (F / (math.pi * (l / F))))
	elif F <= 5e-149:
		tmp = (math.pi * l) - ((F * F) / t_0)
	else:
		tmp = (math.pi * l) - (t_0 / (F * F))
	return tmp

Julia

F = abs(F)
function code(F, l)
	t_0 = tan(Float64(pi * l))
	tmp = 0.0
	if (F <= 2.25e-197)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(F / Float64(pi * Float64(l / F)))));
	elseif (F <= 5e-149)
		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

F = abs(F)
function tmp_2 = code(F, l)
	t_0 = tan((pi * l));
	tmp = 0.0;
	if (F <= 2.25e-197)
		tmp = (pi * l) + (-1.0 / (F / (pi * (l / F))));
	elseif (F <= 5e-149)
		tmp = (pi * l) - ((F * F) / t_0);
	else
		tmp = (pi * l) - (t_0 / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := Block[{t$95$0 = N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, 2.25e-197], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F / N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-149], 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 3 regimes

2. if F < 2.25e-197

   1. Initial program 71.7%

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

   2. Taylor expanded in l around 0 67.0%

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

      1. unpow2 67.0%

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

      2. times-frac 73.7%

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

   4. Simplified 73.7%

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

      1. associate-*r/ 73.7%

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

      2. clear-num 73.7%

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

      3. *-commutative 73.7%

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

   6. Applied egg-rr 73.7%

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

   if 2.25e-197 < F < 4.99999999999999968e-149

   1. Initial program 18.1%

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

      1. inv-pow 18.1%

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

      2. unpow-prod-down 18.1%

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

      3. inv-pow 18.1%

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

      4. inv-pow 18.1%

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

   3. Applied egg-rr 18.1%

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

      1. un-div-inv 18.1%

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

      2. clear-num 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. *-commutative 18.1%

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

      2. associate-/r/ 18.1%

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

      3. associate-*r* 32.9%

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

      4. div-inv 32.9%

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

      5. div-inv 32.9%

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

      6. clear-num 32.9%

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

      7. unpow-1 32.9%

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

      8. exp-to-pow 9.2%

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

      9. add-sqr-sqrt 9.2%

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

      10. sqrt-unprod 9.2%

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

      11. *-commutative 9.2%

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

      12. *-commutative 9.2%

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

   7. Applied egg-rr 69.8%

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

   if 4.99999999999999968e-149 < F

   1. Initial program 89.9%

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

      1. associate-*l/ 90.0%

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

      2. *-lft-identity 90.0%

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

   3. Simplified 90.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.25 \cdot 10^{-197}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\pi \cdot \frac{\ell}{F}}}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\pi \cdot \ell - \frac{F \cdot F}{\tan \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 2: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) F)))

C

F = abs(F);
double code(double F, double l) {
	return (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F) / F);
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	return (Math.PI * l) - ((Math.tan((l * Math.cbrt(Math.pow(Math.PI, 3.0)))) / F) / F);
}

Julia

F = abs(F)
function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F) / F))
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. associate-*l/ 76.2%

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

   2. *-un-lft-identity 76.2%

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

   3. associate-/r* 80.6%

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

3. Applied egg-rr 80.6%

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

   1. add-cbrt-cube 80.6%

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

   2. pow3 80.6%

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

5. Applied egg-rr 80.6%

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

6. Final simplification 80.6%

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

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\pi \cdot \frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-165)
   (+ (* PI l) (/ -1.0 (/ F (* PI (/ l F)))))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-165) {
		tmp = (((double) M_PI) * l) + (-1.0 / (F / (((double) M_PI) * (l / F))));
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-165) {
		tmp = (Math.PI * l) + (-1.0 / (F / (Math.PI * (l / F))));
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if F <= 1.1e-165:
		tmp = (math.pi * l) + (-1.0 / (F / (math.pi * (l / F))))
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (F <= 1.1e-165)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(F / Float64(pi * Float64(l / F)))));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.1e-165)
		tmp = (pi * l) + (-1.0 / (F / (pi * (l / F))));
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[F, 1.1e-165], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F / N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision]), $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 F < 1.0999999999999999e-165

   1. Initial program 69.1%

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

   2. Taylor expanded in l around 0 63.9%

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

      1. unpow2 63.9%

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

      2. times-frac 71.6%

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

   4. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. clear-num 71.6%

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

      3. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

   if 1.0999999999999999e-165 < F

   1. Initial program 86.0%

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

      1. associate-*l/ 86.1%

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

      2. *-lft-identity 86.1%

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

   3. Simplified 86.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\pi \cdot \frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 4: 82.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (tan (* PI l)) F) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. associate-*l/ 76.2%

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

   2. *-un-lft-identity 76.2%

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

   3. associate-/r* 80.6%

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

3. Applied egg-rr 80.6%

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

4. Final simplification 80.6%

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

Alternative 5: 74.9% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (* (/ l F) (/ PI F))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

2. Taylor expanded in l around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 73.1%

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

4. Simplified 73.1%

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

5. Final simplification 73.1%

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

Alternative 6: 74.9% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (* PI (/ l F)) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. associate-*l/ 76.2%

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

   2. *-un-lft-identity 76.2%

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

   3. associate-/r* 80.6%

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

3. Applied egg-rr 80.6%

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

4. Taylor expanded in l around 0 73.1%

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

   1. *-commutative 73.1%

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

   2. associate-*r/ 73.1%

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

6. Simplified 73.1%

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

7. Final simplification 73.1%

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

Alternative 7: 74.9% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ PI F) (/ F l))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] / N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

2. Taylor expanded in l around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 73.1%

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

4. Simplified 73.1%

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

   1. *-commutative 73.1%

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

   2. clear-num 73.0%

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

   3. un-div-inv 73.1%

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

6. Applied egg-rr 73.1%

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

7. Final simplification 73.1%

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

Alternative 8: 74.8% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (* PI l) F) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. associate-*l/ 76.2%

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

   2. *-un-lft-identity 76.2%

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

   3. associate-/r* 80.6%

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

3. Applied egg-rr 80.6%

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

4. Taylor expanded in l around 0 73.1%

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

5. Final simplification 73.1%

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

Reproduce

herbie shell --seed 2023223 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))