VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 98.2%

Time: 19.2s

Alternatives: 7

Speedup: 7.1×

• could not determine a ground truth

  1. F = -18599558.928529024
  2. l = -1547295570780294.5

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: 76.5% 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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.1:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 0.1)
    (+ (* PI l_m) (/ -1.0 (/ F (/ (tan (* PI l_m)) F))))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.1) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F / (tan((((double) M_PI) * l_m)) / F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 0.1) {
		tmp = (Math.PI * l_m) + (-1.0 / (F / (Math.tan((Math.PI * l_m)) / F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 0.1:
		tmp = (math.pi * l_m) + (-1.0 / (F / (math.tan((math.pi * l_m)) / F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.1)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(tan(Float64(pi * l_m)) / F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 0.1)
		tmp = (pi * l_m) + (-1.0 / (F / (tan((pi * l_m)) / F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.1], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 0.10000000000000001

   1. Initial program 79.9%

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

      1. associate-*l/ 80.4%

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

      2. *-un-lft-identity 80.4%

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

      3. associate-/r* 85.6%

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

      4. clear-num 85.6%

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

   4. Applied egg-rr 85.6%

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

   if 0.10000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 64.6%

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

      1. associate-*l/ 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-/r* 64.6%

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

      4. add-sqr-sqrt 25.0%

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

      5. associate-/r* 25.0%

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

   4. Applied egg-rr 25.0%

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

   5. Taylor expanded in l around inf 99.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.1:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.1:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 0.1)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.1) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 0.1) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 0.1:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.1)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 0.1)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.1], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 0.10000000000000001

   1. Initial program 79.9%

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

      1. associate-*l/ 80.4%

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

      2. *-un-lft-identity 80.4%

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

      3. associate-/r* 85.6%

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

   4. Applied egg-rr 85.6%

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

   if 0.10000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 64.6%

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

      1. associate-*l/ 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-/r* 64.6%

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

      4. add-sqr-sqrt 25.0%

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

      5. associate-/r* 25.0%

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

   4. Applied egg-rr 25.0%

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

   5. Taylor expanded in l around inf 99.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.1:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m + l\_m \cdot \frac{\pi}{F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.1:\\ \;\;\;\;t\_0 - \frac{t\_0}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (+ (* PI l_m) (* l_m (/ PI F)))))
   (* l_s (if (<= (* PI l_m) 0.1) (- t_0 (/ t_0 F)) (* PI l_m)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = (((double) M_PI) * l_m) + (l_m * (((double) M_PI) / F));
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.1) {
		tmp = t_0 - (t_0 / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = (Math.PI * l_m) + (l_m * (Math.PI / F));
	double tmp;
	if ((Math.PI * l_m) <= 0.1) {
		tmp = t_0 - (t_0 / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = (math.pi * l_m) + (l_m * (math.pi / F))
	tmp = 0
	if (math.pi * l_m) <= 0.1:
		tmp = t_0 - (t_0 / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(pi * l_m) + Float64(l_m * Float64(pi / F)))
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.1)
		tmp = Float64(t_0 - Float64(t_0 / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	t_0 = (pi * l_m) + (l_m * (pi / F));
	tmp = 0.0;
	if ((pi * l_m) <= 0.1)
		tmp = t_0 - (t_0 / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[(N[(Pi * l$95$m), $MachinePrecision] + N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.1], N[(t$95$0 - N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 0.10000000000000001

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. sqr-neg 79.9%

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

      3. associate-*r/ 80.4%

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

      4. sqr-neg 80.4%

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

      5. *-rgt-identity 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in l around 0 75.9%

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

      1. add-cbrt-cube 75.6%

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

      2. pow3 75.6%

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

   7. Applied egg-rr 75.6%

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

      1. rem-cbrt-cube 75.9%

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

      2. *-commutative 75.9%

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

      3. add-sqr-sqrt 30.2%

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

      4. add-sqr-sqrt 30.2%

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

      5. difference-of-squares 30.2%

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

      6. sqrt-div 30.2%

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

      7. sqrt-prod 16.4%

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

      8. add-sqr-sqrt 22.8%

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

      9. sqrt-div 22.8%

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

      10. sqrt-prod 17.8%

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

      11. add-sqr-sqrt 32.5%

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

   9. Applied egg-rr 32.5%

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

       1. sub-neg 32.5%

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

       2. distribute-rgt-in 32.5%

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

       3. distribute-lft-in 32.5%

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

       4. rem-square-sqrt 32.6%

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

       5. associate-*r/ 32.6%

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

       6. rem-square-sqrt 32.6%

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

       7. associate-*r/ 32.6%

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

       8. distribute-neg-frac2 32.6%

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

   11. Simplified 81.1%

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

   if 0.10000000000000001 < (*.f64 (PI.f64) l)

   1. Initial program 64.6%

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

      1. associate-*l/ 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-/r* 64.6%

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

      4. add-sqr-sqrt 25.0%

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

      5. associate-/r* 25.0%

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

   4. Applied egg-rr 25.0%

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

   5. Taylor expanded in l around inf 99.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.1:\\ \;\;\;\;\left(\pi \cdot \ell + \ell \cdot \frac{\pi}{F}\right) - \frac{\pi \cdot \ell + \ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 5.3 \cdot 10^{-300}:\\ \;\;\;\;\pi \cdot \left(\frac{l\_m}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-243} \lor \neg \left(F \leq 1.1 \cdot 10^{-91}\right):\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot l\_m}{F} \cdot \frac{-1}{F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= F 5.3e-300)
    (* PI (* (/ l_m F) (/ -1.0 F)))
    (if (or (<= F 1.2e-243) (not (<= F 1.1e-91)))
      (* PI l_m)
      (* (/ (* PI l_m) F) (/ -1.0 F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (F <= 5.3e-300) {
		tmp = ((double) M_PI) * ((l_m / F) * (-1.0 / F));
	} else if ((F <= 1.2e-243) || !(F <= 1.1e-91)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = ((((double) M_PI) * l_m) / F) * (-1.0 / F);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (F <= 5.3e-300) {
		tmp = Math.PI * ((l_m / F) * (-1.0 / F));
	} else if ((F <= 1.2e-243) || !(F <= 1.1e-91)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = ((Math.PI * l_m) / F) * (-1.0 / F);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if F <= 5.3e-300:
		tmp = math.pi * ((l_m / F) * (-1.0 / F))
	elif (F <= 1.2e-243) or not (F <= 1.1e-91):
		tmp = math.pi * l_m
	else:
		tmp = ((math.pi * l_m) / F) * (-1.0 / F)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (F <= 5.3e-300)
		tmp = Float64(pi * Float64(Float64(l_m / F) * Float64(-1.0 / F)));
	elseif ((F <= 1.2e-243) || !(F <= 1.1e-91))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(Float64(pi * l_m) / F) * Float64(-1.0 / F));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (F <= 5.3e-300)
		tmp = pi * ((l_m / F) * (-1.0 / F));
	elseif ((F <= 1.2e-243) || ~((F <= 1.1e-91)))
		tmp = pi * l_m;
	else
		tmp = ((pi * l_m) / F) * (-1.0 / F);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[F, 5.3e-300], N[(Pi * N[(N[(l$95$m / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.2e-243], N[Not[LessEqual[F, 1.1e-91]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if F < 5.2999999999999999e-300

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

      2. sqr-neg 73.5%

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

      3. associate-*r/ 73.8%

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

      4. sqr-neg 73.8%

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

      5. *-rgt-identity 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in l around 0 66.7%

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

      1. add-exp-log 31.7%

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

   7. Applied egg-rr 31.7%

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

   8. Taylor expanded in F around 0 23.4%

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

      1. mul-1-neg 23.4%

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

      2. *-commutative 23.4%

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

      3. associate-*r/ 23.5%

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

      4. distribute-lft-neg-in 23.5%

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

   10. Simplified 23.5%

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

       1. *-un-lft-identity 23.5%

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

       2. pow2 23.5%

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

       3. times-frac 28.4%

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

   12. Applied egg-rr 28.4%

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

   if 5.2999999999999999e-300 < F < 1.2e-243 or 1.1e-91 < F

   1. Initial program 84.4%

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

      1. associate-*l/ 84.4%

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

      2. *-un-lft-identity 84.4%

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

      3. associate-/r* 84.4%

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

      4. add-sqr-sqrt 84.3%

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

      5. associate-/r* 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in l around inf 85.4%

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

   if 1.2e-243 < F < 1.1e-91

   1. Initial program 63.5%

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

   3. Taylor expanded in F around 0 63.0%

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

      1. mul-1-neg 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. *-commutative 63.0%

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

      5. distribute-frac-neg2 63.0%

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

      6. distribute-rgt-neg-out 63.0%

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

      7. *-commutative 63.0%

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

      8. *-commutative 63.0%

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

      9. distribute-lft-neg-out 63.0%

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

      10. distribute-rgt-neg-in 63.0%

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

      11. *-commutative 63.0%

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

   5. Simplified 63.0%

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

      1. pow2 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/r* 62.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 1.3%

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

      6. sqr-neg 1.3%

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

      7. sqrt-unprod 1.3%

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

      8. *-commutative 1.3%

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

      9. add-sqr-sqrt 1.3%

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

      10. *-commutative 1.3%

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

      11. tan-quot 1.3%

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

      12. associate-/l/ 1.3%

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

   7. Applied egg-rr 74.3%

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

      1. *-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in l around 0 72.9%

       \[\leadsto \frac{\color{blue}{\ell \cdot \pi}}{F} \cdot \frac{1}{-F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.3 \cdot 10^{-300}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-243} \lor \neg \left(F \leq 1.1 \cdot 10^{-91}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F} \cdot \frac{-1}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 1.6 \cdot 10^{-299} \lor \neg \left(F \leq 3.6 \cdot 10^{-242}\right) \land F \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\pi \cdot \left(\frac{l\_m}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= F 1.6e-299) (and (not (<= F 3.6e-242)) (<= F 1.85e-91)))
    (* PI (* (/ l_m F) (/ -1.0 F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((F <= 1.6e-299) || (!(F <= 3.6e-242) && (F <= 1.85e-91))) {
		tmp = ((double) M_PI) * ((l_m / F) * (-1.0 / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((F <= 1.6e-299) || (!(F <= 3.6e-242) && (F <= 1.85e-91))) {
		tmp = Math.PI * ((l_m / F) * (-1.0 / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (F <= 1.6e-299) or (not (F <= 3.6e-242) and (F <= 1.85e-91)):
		tmp = math.pi * ((l_m / F) * (-1.0 / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((F <= 1.6e-299) || (!(F <= 3.6e-242) && (F <= 1.85e-91)))
		tmp = Float64(pi * Float64(Float64(l_m / F) * Float64(-1.0 / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((F <= 1.6e-299) || (~((F <= 3.6e-242)) && (F <= 1.85e-91)))
		tmp = pi * ((l_m / F) * (-1.0 / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[Or[LessEqual[F, 1.6e-299], And[N[Not[LessEqual[F, 3.6e-242]], $MachinePrecision], LessEqual[F, 1.85e-91]]], N[(Pi * N[(N[(l$95$m / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 1.60000000000000004e-299 or 3.60000000000000014e-242 < F < 1.8500000000000001e-91

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

      2. sqr-neg 71.9%

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

      3. associate-*r/ 72.5%

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

      4. sqr-neg 72.5%

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

      5. *-rgt-identity 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in l around 0 65.8%

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

      1. add-exp-log 31.5%

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

   7. Applied egg-rr 31.5%

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

   8. Taylor expanded in F around 0 29.6%

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

      1. mul-1-neg 29.6%

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

      2. *-commutative 29.6%

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

      3. associate-*r/ 29.7%

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

      4. distribute-lft-neg-in 29.7%

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

   10. Simplified 29.7%

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

       1. *-un-lft-identity 29.7%

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

       2. pow2 29.7%

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

       3. times-frac 35.6%

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

   12. Applied egg-rr 35.6%

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

   if 1.60000000000000004e-299 < F < 3.60000000000000014e-242 or 1.8500000000000001e-91 < F

   1. Initial program 84.4%

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

      1. associate-*l/ 84.4%

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

      2. *-un-lft-identity 84.4%

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

      3. associate-/r* 84.4%

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

      4. add-sqr-sqrt 84.3%

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

      5. associate-/r* 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in l around inf 85.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.6 \cdot 10^{-299} \lor \neg \left(F \leq 3.6 \cdot 10^{-242}\right) \land F \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 0.5) (- (* PI l_m) (* (/ PI F) (/ l_m F))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 0.5:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 0.5)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 0.5)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 0.5], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. sqr-neg 79.9%

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

      3. associate-*r/ 80.4%

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

      4. sqr-neg 80.4%

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

      5. *-rgt-identity 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in l around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 81.1%

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

   7. Applied egg-rr 81.1%

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

   if 0.5 < l

   1. Initial program 64.6%

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

      1. associate-*l/ 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-/r* 64.6%

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

      4. add-sqr-sqrt 25.0%

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

      5. associate-/r* 25.0%

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

   4. Applied egg-rr 25.0%

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

   5. Taylor expanded in l around inf 99.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* PI l_m)))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * (((double) M_PI) * l_m);
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * (Math.PI * l_m);
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * (math.pi * l_m)

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(pi * l_m))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * (pi * l_m);
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. associate-*l/ 76.6%

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

   2. *-un-lft-identity 76.6%

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

   3. associate-/r* 80.6%

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

   4. add-sqr-sqrt 38.7%

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

   5. associate-/r* 38.7%

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

4. Applied egg-rr 38.7%

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

5. Taylor expanded in l around inf 72.4%

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

6. Final simplification 72.4%

   \[\leadsto \pi \cdot \ell \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))