VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.0% → 92.3%

Time: 2.7s

Alternatives: 4

Speedup: 2.6×

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: 76.0% 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: 92.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -6.5e+20)
   (* l PI)
   (if (<= l 0.5) (- (* PI l) (/ (* l PI) (pow F 2.0))) (* l PI))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -6.5e+20) {
		tmp = l * ((double) M_PI);
	} else if (l <= 0.5) {
		tmp = (((double) M_PI) * l) - ((l * ((double) M_PI)) / pow(F, 2.0));
	} else {
		tmp = l * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -6.5e+20) {
		tmp = l * Math.PI;
	} else if (l <= 0.5) {
		tmp = (Math.PI * l) - ((l * Math.PI) / Math.pow(F, 2.0));
	} else {
		tmp = l * Math.PI;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -6.5e+20:
		tmp = l * math.pi
	elif l <= 0.5:
		tmp = (math.pi * l) - ((l * math.pi) / math.pow(F, 2.0))
	else:
		tmp = l * math.pi
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -6.5e+20)
		tmp = Float64(l * pi);
	elseif (l <= 0.5)
		tmp = Float64(Float64(pi * l) - Float64(Float64(l * pi) / (F ^ 2.0)));
	else
		tmp = Float64(l * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -6.5e+20)
		tmp = l * pi;
	elseif (l <= 0.5)
		tmp = (pi * l) - ((l * pi) / (F ^ 2.0));
	else
		tmp = l * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -6.5e+20], N[(l * Pi), $MachinePrecision], If[LessEqual[l, 0.5], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l * Pi), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.5e20 or 0.5 < l

   1. Initial program 76.0%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   4. Applied rewrites 73.3%

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

   if -6.5e20 < l < 0.5

   1. Initial program 76.0%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 68.9%

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

Alternative 2: 91.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\ell \leq 0.5:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -6.5e+20)
   (* l PI)
   (if (<= l 0.5) (* l (- PI (/ PI (* F F)))) (* l PI))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -6.5e+20) {
		tmp = l * ((double) M_PI);
	} else if (l <= 0.5) {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = l * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -6.5e+20) {
		tmp = l * Math.PI;
	} else if (l <= 0.5) {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = l * Math.PI;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -6.5e+20:
		tmp = l * math.pi
	elif l <= 0.5:
		tmp = l * (math.pi - (math.pi / (F * F)))
	else:
		tmp = l * math.pi
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -6.5e+20)
		tmp = Float64(l * pi);
	elseif (l <= 0.5)
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(l * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -6.5e+20)
		tmp = l * pi;
	elseif (l <= 0.5)
		tmp = l * (pi - (pi / (F * F)));
	else
		tmp = l * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -6.5e+20], N[(l * Pi), $MachinePrecision], If[LessEqual[l, 0.5], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.5e20 or 0.5 < l

   1. Initial program 76.0%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   4. Applied rewrites 73.3%

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

   if -6.5e20 < l < 0.5

   1. Initial program 76.0%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 44.0%

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

   9. Applied rewrites 68.5%

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

Alternative 3: 73.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-22}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-293}:\\ \;\;\;\;-\frac{\ell \cdot \pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= l -7e-22)
   (* l PI)
   (if (<= l -2.05e-293) (- (/ (* l PI) (* F F))) (* l PI))))

C

double code(double F, double l) {
	double tmp;
	if (l <= -7e-22) {
		tmp = l * ((double) M_PI);
	} else if (l <= -2.05e-293) {
		tmp = -((l * ((double) M_PI)) / (F * F));
	} else {
		tmp = l * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (l <= -7e-22) {
		tmp = l * Math.PI;
	} else if (l <= -2.05e-293) {
		tmp = -((l * Math.PI) / (F * F));
	} else {
		tmp = l * Math.PI;
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if l <= -7e-22:
		tmp = l * math.pi
	elif l <= -2.05e-293:
		tmp = -((l * math.pi) / (F * F))
	else:
		tmp = l * math.pi
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (l <= -7e-22)
		tmp = Float64(l * pi);
	elseif (l <= -2.05e-293)
		tmp = Float64(-Float64(Float64(l * pi) / Float64(F * F)));
	else
		tmp = Float64(l * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (l <= -7e-22)
		tmp = l * pi;
	elseif (l <= -2.05e-293)
		tmp = -((l * pi) / (F * F));
	else
		tmp = l * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[l, -7e-22], N[(l * Pi), $MachinePrecision], If[LessEqual[l, -2.05e-293], (-N[(N[(l * Pi), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), N[(l * Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -7.00000000000000011e-22 or -2.04999999999999994e-293 < l

   1. Initial program 76.0%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   4. Applied rewrites 73.3%

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

   if -7.00000000000000011e-22 < l < -2.04999999999999994e-293

   1. Initial program 76.0%

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

   2. Taylor expanded in l around 0

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 21.3%

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

Alternative 4: 71.6% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 73.3%

   \[\leadsto \color{blue}{\ell \cdot \pi} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025160 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))