VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.3%

Time: 3.9s

Alternatives: 8

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: 76.9% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{-\frac{\sin \left(\pi \cdot l\_m\right) \cdot {F}^{-1}}{F}}{\cos \left(\mathsf{fma}\left(\pi, l\_m, \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 2.25e+15)
    (-
     (* PI l_m)
     (/ (- (/ (* (sin (* PI l_m)) (pow F -1.0)) F)) (cos (fma PI l_m PI))))
    (* 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 <= 2.25e+15) {
		tmp = (((double) M_PI) * l_m) - (-((sin((((double) M_PI) * l_m)) * pow(F, -1.0)) / F) / cos(fma(((double) M_PI), l_m, ((double) M_PI))));
	} else {
		tmp = ((double) M_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 <= 2.25e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(-Float64(Float64(sin(Float64(pi * l_m)) * (F ^ -1.0)) / F)) / cos(fma(pi, l_m, pi))));
	else
		tmp = Float64(pi * l_m);
	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, 2.25e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[((-N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[Power[F, -1.0], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]) / N[Cos[N[(Pi * l$95$m + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.25e15

   1. Initial program 87.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. 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 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. *-commutative N/A

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

      13. frac-2neg N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   if 2.25e15 < l

   1. Initial program 64.7%

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

   2. Taylor expanded in F around inf

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

      1. *-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 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(-\pi\right) \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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)))) -5e-168)
    (/ (* (- 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)))) <= -5e-168) {
		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)))) <= -5e-168) {
		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)))) <= -5e-168:
		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)))) <= -5e-168)
		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)))) <= -5e-168)
		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], -5e-168], 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)))) < -5.00000000000000001e-168

   1. Initial program 56.8%

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

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   3. 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 56.2

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   6. 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-/l* 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. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if -5.00000000000000001e-168 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 87.1%

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

   2. Taylor expanded in F around inf

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

      1. *-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 96.3

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

   4. Applied rewrites 96.3%

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

Alternative 3: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -5 \cdot 10^{-168}:\\ \;\;\;\;-l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -5e-168)
    (- (* 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 (((((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -5e-168) {
		tmp = -(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 (((Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -5e-168) {
		tmp = -(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 ((math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -5e-168:
		tmp = -(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 (Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -5e-168)
		tmp = Float64(-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 (((pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -5e-168)
		tmp = -(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[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], -5e-168], (-N[(l$95$m * N[(Pi / N[(F * F), $MachinePrecision]), $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)))) < -5.00000000000000001e-168

   1. Initial program 56.8%

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

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   3. 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 56.2

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   6. 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-/l* 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. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if -5.00000000000000001e-168 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 87.1%

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

   2. Taylor expanded in F around inf

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

      1. *-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 96.3

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

   4. Applied rewrites 96.3%

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

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{-1}{F} \cdot \tan \left(\pi \cdot l\_m\right)}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4.4e+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 <= 4.4e+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 <= 4.4e+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 <= 4.4e+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 <= 4.4e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(-1.0 / F) * tan(Float64(pi * l_m))) / Float64(-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 <= 4.4e+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, 4.4e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(-1.0 / F), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e15

   1. Initial program 87.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. 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 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-pow.f64 N/A

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

      12. inv-pow N/A

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

      13. associate-/r* N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

   if 4.4e15 < l

   1. Initial program 64.7%

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

   2. Taylor expanded in F around inf

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

      1. *-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 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 5: 98.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{-\frac{\pi \cdot l\_m}{F}}{-F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 3.7e+15)
    (- (* 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 <= 3.7e+15) {
		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 <= 3.7e+15) {
		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 <= 3.7e+15:
		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 <= 3.7e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(-Float64(Float64(pi * l_m) / F)) / Float64(-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 <= 3.7e+15)
		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, 3.7e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[((-N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision]) / (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.7e15

   1. Initial program 87.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. 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 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-pow.f64 N/A

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

      12. inv-pow N/A

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

      13. associate-/r* N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative 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. lift-PI.f64 97.5

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

   8. Applied rewrites 97.5%

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

   if 3.7e15 < l

   1. Initial program 64.7%

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

   2. Taylor expanded in F around inf

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

      1. *-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 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 6: 92.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 3.7e+15) (* (- 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 <= 3.7e+15) {
		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 <= 3.7e+15) {
		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 <= 3.7e+15:
		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 <= 3.7e+15)
		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 <= 3.7e+15)
		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, 3.7e+15], 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 < 3.7e15

   1. Initial program 87.6%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   3. 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 86.0

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

   4. Applied rewrites 86.0%

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

   if 3.7e15 < l

   1. Initial program 64.7%

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

   2. Taylor expanded in F around inf

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

      1. *-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 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 7: 72.5% 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 76.9%

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

2. Taylor expanded in F around inf

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

   1. *-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 72.5

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

4. Applied rewrites 72.5%

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

Alternative 8: 3.1% accurate, 135.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot 0 \end{array} \]

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 0.0))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Fortran

l\_m =     private
l\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_s, f, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_s
    real(8), intent (in) :: f
    real(8), intent (in) :: l_m
    code = l_s * 0.0d0
end function

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 * 0.0;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * 0.0

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * 0.0)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
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. 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 83.0

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

3. Applied rewrites 83.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-pow.f64 N/A

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

   12. inv-pow N/A

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

   13. frac-2neg-rev N/A

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

   14. *-commutative N/A

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in l around 0

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

   1. lower-/.f64 N/A

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

   2. sin-PI N/A

      \[\leadsto \frac{0}{{\color{blue}{F}}^{2}} \]

   3. pow2 N/A

      \[\leadsto \frac{0}{F \cdot \color{blue}{F}} \]

   4. lift-*.f64 2.6

      \[\leadsto \frac{0}{F \cdot \color{blue}{F}} \]

8. Applied rewrites 2.6%

   \[\leadsto \color{blue}{\frac{0}{F \cdot F}} \]

9. Taylor expanded in F around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 3.1%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))