VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 99.3%

Time: 3.6s

Alternatives: 8

Speedup: 3.2×

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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.36 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \tan \left(l\_m \cdot \pi\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.36e+15)
    (- (* PI l_m) (/ (* (/ 1.0 F) (tan (* l_m PI))) F))
    (* l_m PI))))

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 <= 1.36e+15) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * tan((l_m * ((double) M_PI)))) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.36e+15) {
		tmp = (Math.PI * l_m) - (((1.0 / F) * Math.tan((l_m * Math.PI))) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.36e+15:
		tmp = (math.pi * l_m) - (((1.0 / F) * math.tan((l_m * math.pi))) / F)
	else:
		tmp = l_m * math.pi
	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 <= 1.36e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * tan(Float64(l_m * pi))) / F));
	else
		tmp = Float64(l_m * pi);
	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 <= 1.36e+15)
		tmp = (pi * l_m) - (((1.0 / F) * tan((l_m * pi))) / F);
	else
		tmp = l_m * pi;
	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, 1.36e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.36e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if 1.36e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.36 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(l\_m \cdot \pi\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.36e+15)
    (- (* PI l_m) (/ (/ (tan (* l_m PI)) F) F))
    (* l_m PI))))

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 <= 1.36e+15) {
		tmp = (((double) M_PI) * l_m) - ((tan((l_m * ((double) M_PI))) / F) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.36e+15) {
		tmp = (Math.PI * l_m) - ((Math.tan((l_m * Math.PI)) / F) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.36e+15:
		tmp = (math.pi * l_m) - ((math.tan((l_m * math.pi)) / F) / F)
	else:
		tmp = l_m * math.pi
	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 <= 1.36e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(l_m * pi)) / F) / F));
	else
		tmp = Float64(l_m * pi);
	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 <= 1.36e+15)
		tmp = (pi * l_m) - ((tan((l_m * pi)) / F) / F);
	else
		tmp = l_m * pi;
	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, 1.36e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.36e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

   if 1.36e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 3: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\mathsf{fma}\left(\left(l\_m \cdot l\_m\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \frac{\pi}{F}\right), 0.3333333333333333, \frac{\pi}{F}\right) \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.3e+15)
    (-
     (* PI l_m)
     (/
      (*
       (fma (* (* l_m l_m) (* (* PI PI) (/ PI F))) 0.3333333333333333 (/ PI F))
       l_m)
      F))
    (* l_m PI))))

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 <= 1.3e+15) {
		tmp = (((double) M_PI) * l_m) - ((fma(((l_m * l_m) * ((((double) M_PI) * ((double) M_PI)) * (((double) M_PI) / F))), 0.3333333333333333, (((double) M_PI) / F)) * l_m) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.3e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(fma(Float64(Float64(l_m * l_m) * Float64(Float64(pi * pi) * Float64(pi / F))), 0.3333333333333333, Float64(pi / F)) * l_m) / F));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(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, 1.3e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(Pi / F), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

   4. Taylor expanded in l around 0

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right) + \frac{\mathsf{PI}\left(\right)}{F}\right)}}{F} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right) + \frac{\mathsf{PI}\left(\right)}{F}\right)}}{F} \]

      2. lower-fma.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\ell \cdot \mathsf{fma}\left({\ell}^{2}, \color{blue}{\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}}, \frac{\mathsf{PI}\left(\right)}{F}\right)}{F} \]

   6. Applied rewrites 59.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \mathsf{fma}\left({\ell}^{2}, -0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}, \frac{\pi}{F}\right)}}{F} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-1}{6} \cdot \frac{{\pi}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\pi}^{3}}{F}, \frac{\pi}{F}\right)}}{F} \]

      2. *-commutative N/A

         \[\leadsto \pi \cdot \ell - \frac{\mathsf{fma}\left({\ell}^{2}, \frac{-1}{6} \cdot \frac{{\pi}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\pi}^{3}}{F}, \frac{\pi}{F}\right) \cdot \color{blue}{\ell}}{F} \]

      3. lower-*.f64 59.5

         \[\leadsto \pi \cdot \ell - \frac{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}, \frac{\pi}{F}\right) \cdot \color{blue}{\ell}}{F} \]

   8. Applied rewrites 59.6%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \frac{\pi}{F}\right), 0.3333333333333333, \frac{\pi}{F}\right) \cdot \ell}}{F} \]

   if 1.3e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 4: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\pi}{F} \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.3e+15) (- (* PI l_m) (/ (* (/ PI F) l_m) F)) (* l_m PI))))

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 <= 1.3e+15) {
		tmp = (((double) M_PI) * l_m) - (((((double) M_PI) / F) * l_m) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.3e+15) {
		tmp = (Math.PI * l_m) - (((Math.PI / F) * l_m) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.3e+15:
		tmp = (math.pi * l_m) - (((math.pi / F) * l_m) / F)
	else:
		tmp = l_m * math.pi
	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 <= 1.3e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi / F) * l_m) / F));
	else
		tmp = Float64(l_m * pi);
	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 <= 1.3e+15)
		tmp = (pi * l_m) - (((pi / F) * l_m) / F);
	else
		tmp = l_m * pi;
	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, 1.3e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi / F), $MachinePrecision] * l$95$m), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 74.9

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

   8. Applied rewrites 74.9%

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

   if 1.3e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 5: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.3e+15) (- (* PI l_m) (/ (* PI (/ l_m F)) F)) (* l_m PI))))

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 <= 1.3e+15) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.3e+15) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.3e+15:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = l_m * math.pi
	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 <= 1.3e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(l_m * pi);
	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 <= 1.3e+15)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = l_m * pi;
	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, 1.3e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

   if 1.3e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 6: 92.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\frac{\pi}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.3e+15) (* l_m (- PI (/ (/ PI F) F))) (* l_m PI))))

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 <= 1.3e+15) {
		tmp = l_m * (((double) M_PI) - ((((double) M_PI) / F) / F));
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.3e+15) {
		tmp = l_m * (Math.PI - ((Math.PI / F) / F));
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.3e+15:
		tmp = l_m * (math.pi - ((math.pi / F) / F))
	else:
		tmp = l_m * math.pi
	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 <= 1.3e+15)
		tmp = Float64(l_m * Float64(pi - Float64(Float64(pi / F) / F)));
	else
		tmp = Float64(l_m * pi);
	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 <= 1.3e+15)
		tmp = l_m * (pi - ((pi / F) / F));
	else
		tmp = l_m * pi;
	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, 1.3e+15], N[(l$95$m * N[(Pi - N[(N[(Pi / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-pow.f64 68.9

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

   8. Applied rewrites 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if 1.3e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 7: 92.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

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 1.3e+15) (* (- PI (/ PI (* F F))) l_m) (* l_m PI))))

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 <= 1.3e+15) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * l_m;
	} else {
		tmp = l_m * ((double) M_PI);
	}
	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 <= 1.3e+15) {
		tmp = (Math.PI - (Math.PI / (F * F))) * l_m;
	} else {
		tmp = l_m * Math.PI;
	}
	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 <= 1.3e+15:
		tmp = (math.pi - (math.pi / (F * F))) * l_m
	else:
		tmp = l_m * math.pi
	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 <= 1.3e+15)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * l_m);
	else
		tmp = Float64(l_m * pi);
	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 <= 1.3e+15)
		tmp = (pi - (pi / (F * F))) * l_m;
	else
		tmp = l_m * pi;
	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, 1.3e+15], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.3

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

   3. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-pow.f64 68.9

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

   8. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.9

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if 1.3e15 < l

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.2

         \[\leadsto \ell \cdot \pi \]

   4. Applied rewrites 73.2%

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

Alternative 8: 73.2% accurate, 13.6× speedup

Math

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

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 (* l_m PI)))

C

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

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 * (l_m * Math.PI);
}

Python

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

Julia

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

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * (l_m * pi);
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[(l$95$m * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 73.2

      \[\leadsto \ell \cdot \pi \]

4. Applied rewrites 73.2%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))