VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 99.2%

Time: 3.5s

Alternatives: 10

Speedup: 4.4×

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: 75.9% 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{{F}^{-1} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 1.2e+15)
    (- (* PI l_m) (/ (* (pow F -1.0) (sin (* PI l_m))) (* F (cos (* PI l_m)))))
    (* 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 <= 1.2e+15) {
		tmp = (((double) M_PI) * l_m) - ((pow(F, -1.0) * sin((((double) M_PI) * l_m))) / (F * cos((((double) M_PI) * l_m))));
	} 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 <= 1.2e+15) {
		tmp = (Math.PI * l_m) - ((Math.pow(F, -1.0) * Math.sin((Math.PI * l_m))) / (F * Math.cos((Math.PI * l_m))));
	} 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 <= 1.2e+15:
		tmp = (math.pi * l_m) - ((math.pow(F, -1.0) * math.sin((math.pi * l_m))) / (F * math.cos((math.pi * l_m))))
	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 <= 1.2e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64((F ^ -1.0) * sin(Float64(pi * l_m))) / Float64(F * cos(Float64(pi * l_m)))));
	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 <= 1.2e+15)
		tmp = (pi * l_m) - (((F ^ -1.0) * sin((pi * l_m))) / (F * cos((pi * l_m))));
	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, 1.2e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Power[F, -1.0], $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e15

   1. Initial program 87.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 1.2e15 < l

   1. Initial program 63.7%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 2: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -4 \cdot 10^{-246}:\\ \;\;\;\;\frac{\left(-\pi\right) \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -4e-246)
    (/ (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -4e-246) {
		tmp = (-((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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -4e-246) {
		tmp = (-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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -4e-246:
		tmp = (-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(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -4e-246)
		tmp = Float64(Float64(Float64(-pi) * l_m) / Float64(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) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -4e-246)
		tmp = (-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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-246], N[(N[((-Pi) * l$95$m), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -3.99999999999999982e-246

   1. Initial program 55.9%

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

   3. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 56.0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\left(-1 \cdot \mathsf{PI}\left(\right)\right) \cdot \ell}{F \cdot F} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   if -3.99999999999999982e-246 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 86.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 3: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -4 \cdot 10^{-246}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -4e-246)
    (* (/ PI (* F F)) (- l_m))
    (* 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -4e-246) {
		tmp = (((double) M_PI) / (F * F)) * -l_m;
	} 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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -4e-246) {
		tmp = (Math.PI / (F * F)) * -l_m;
	} 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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -4e-246:
		tmp = (math.pi / (F * F)) * -l_m
	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(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -4e-246)
		tmp = Float64(Float64(pi / Float64(F * F)) * Float64(-l_m));
	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) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -4e-246)
		tmp = (pi / (F * F)) * -l_m;
	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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-246], N[(N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision] * (-l$95$m)), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -3.99999999999999982e-246

   1. Initial program 55.9%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r/ N/A

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

      3. lower-neg.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 54.5

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

   8. Applied rewrites 54.5%

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

   if -3.99999999999999982e-246 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 86.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq -4 \cdot 10^{-246}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{F} \cdot \frac{\tan \left(\pi \cdot l\_m\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 1.2e+15)
    (+ (* 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 (l_m <= 1.2e+15) {
		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 (l_m <= 1.2e+15) {
		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 l_m <= 1.2e+15:
		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 (l_m <= 1.2e+15)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / 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 (l_m <= 1.2e+15)
		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[l$95$m, 1.2e+15], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] * N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e15

   1. Initial program 87.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 1.2e15 < l

   1. Initial program 63.7%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4400000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 4400000000000.0)
    (- (* PI l_m) (/ (* l_m (/ PI 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 (l_m <= 4400000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / 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 (l_m <= 4400000000000.0) {
		tmp = (Math.PI * l_m) - ((l_m * (Math.PI / 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 l_m <= 4400000000000.0:
		tmp = (math.pi * l_m) - ((l_m * (math.pi / 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 (l_m <= 4400000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / 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 (l_m <= 4400000000000.0)
		tmp = (pi * l_m) - ((l_m * (pi / 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[l$95$m, 4400000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e12

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 97.8

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

   9. Applied rewrites 97.8%

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

   if 4.4e12 < l

   1. Initial program 63.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 6: 98.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4400000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4400000000000.0)
    (- (* 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 <= 4400000000000.0) {
		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 <= 4400000000000.0) {
		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 <= 4400000000000.0:
		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 <= 4400000000000.0)
		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 <= 4400000000000.0)
		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, 4400000000000.0], 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 < 4.4e12

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 97.8

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

   9. Applied rewrites 97.8%

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

   if 4.4e12 < l

   1. Initial program 63.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 7: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4400000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4400000000000.0)
    (- (* PI l_m) (/ (* 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 (l_m <= 4400000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((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 (l_m <= 4400000000000.0) {
		tmp = (Math.PI * l_m) - ((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 l_m <= 4400000000000.0:
		tmp = (math.pi * l_m) - ((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 (l_m <= 4400000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * l_m) / Float64(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 (l_m <= 4400000000000.0)
		tmp = (pi * l_m) - ((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[l$95$m, 4400000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * l$95$m), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e12

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if 4.4e12 < l

   1. Initial program 63.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 92.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4400000000000:\\ \;\;\;\;\pi \cdot l\_m - l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4400000000000.0)
    (- (* PI l_m) (* l_m (/ PI (* 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 (l_m <= 4400000000000.0) {
		tmp = (((double) M_PI) * l_m) - (l_m * (((double) M_PI) / (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 (l_m <= 4400000000000.0) {
		tmp = (Math.PI * l_m) - (l_m * (Math.PI / (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 l_m <= 4400000000000.0:
		tmp = (math.pi * l_m) - (l_m * (math.pi / (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 (l_m <= 4400000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(l_m * Float64(pi / Float64(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 (l_m <= 4400000000000.0)
		tmp = (pi * l_m) - (l_m * (pi / (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[l$95$m, 4400000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(l$95$m * N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e12

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 4.4e12 < l

   1. Initial program 63.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 9: 92.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4400000000000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4400000000000.0) (* (- PI (/ PI (* F F))) l_m) (* 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 <= 4400000000000.0) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * l_m;
	} 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 <= 4400000000000.0) {
		tmp = (Math.PI - (Math.PI / (F * F))) * l_m;
	} 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 <= 4400000000000.0:
		tmp = (math.pi - (math.pi / (F * F))) * l_m
	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 <= 4400000000000.0)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * l_m);
	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 <= 4400000000000.0)
		tmp = (pi - (pi / (F * F))) * l_m;
	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, 4400000000000.0], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e12

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 4.4e12 < l

   1. Initial program 63.8%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 10: 72.8% accurate, 22.5× 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 75.9%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 72.8

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

5. Applied rewrites 72.8%

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

Reproduce

herbie shell --seed 2025086 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))