VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 87.2%

Time: 22.6s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. F = -9.808511794946167e+242
  2. l = -3800843246574012.5

Specification

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.8% 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: 87.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \frac{{F_m}^{2}}{{\pi}^{2}}\\ t_2 := 0.008333333333333333 \cdot {\pi}^{5}\\ t_3 := \frac{{F_m}^{2}}{{\pi}^{3}}\\ t_4 := \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_5 := t_4 - t_2\\ t_6 := t_3 \cdot {t_0}^{2} + t_1 \cdot t_5\\ \mathbf{if}\;F_m \leq 1.62 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F_m}}{F_m}\\ \mathbf{elif}\;F_m \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(\left(\frac{{F_m}^{2}}{\pi \cdot \ell} - t_1 \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) - {\ell}^{5} \cdot \left(\left(t_1 \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, t_2 - t_4, \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) + t_3 \cdot \left(t_0 \cdot t_5\right)\right) + t_0 \cdot \frac{t_6}{\pi}\right)\right) + {\ell}^{3} \cdot t_6}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F_m}}{F_m}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1 (/ (pow F_m 2.0) (pow PI 2.0)))
        (t_2 (* 0.008333333333333333 (pow PI 5.0)))
        (t_3 (/ (pow F_m 2.0) (pow PI 3.0)))
        (t_4
         (fma
          -0.5
          (* (pow PI 3.0) (* (pow PI 2.0) 0.3333333333333333))
          (* (pow PI 5.0) 0.041666666666666664)))
        (t_5 (- t_4 t_2))
        (t_6 (+ (* t_3 (pow t_0 2.0)) (* t_1 t_5))))
   (if (<= F_m 1.62e-161)
     (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F_m) F_m))
     (if (<= F_m 5e-52)
       (-
        (* PI l)
        (/
         1.0
         (+
          (-
           (-
            (/ (pow F_m 2.0) (* PI l))
            (* t_1 (* (pow PI 3.0) (* l 0.3333333333333333))))
           (*
            (pow l 5.0)
            (+
             (+
              (*
               t_1
               (-
                (* -0.0001984126984126984 (pow PI 7.0))
                (fma
                 (* (pow PI 2.0) -0.5)
                 (- t_2 t_4)
                 (fma
                  0.041666666666666664
                  (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 4.0)))
                  (* (pow PI 7.0) -0.001388888888888889)))))
              (* t_3 (* t_0 t_5)))
             (* t_0 (/ t_6 PI)))))
          (* (pow l 3.0) t_6))))
       (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F_m) F_m))))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = pow(F_m, 2.0) / pow(((double) M_PI), 2.0);
	double t_2 = 0.008333333333333333 * pow(((double) M_PI), 5.0);
	double t_3 = pow(F_m, 2.0) / pow(((double) M_PI), 3.0);
	double t_4 = fma(-0.5, (pow(((double) M_PI), 3.0) * (pow(((double) M_PI), 2.0) * 0.3333333333333333)), (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_5 = t_4 - t_2;
	double t_6 = (t_3 * pow(t_0, 2.0)) + (t_1 * t_5);
	double tmp;
	if (F_m <= 1.62e-161) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F_m) / F_m);
	} else if (F_m <= 5e-52) {
		tmp = (((double) M_PI) * l) - (1.0 / ((((pow(F_m, 2.0) / (((double) M_PI) * l)) - (t_1 * (pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)))) - (pow(l, 5.0) * (((t_1 * ((-0.0001984126984126984 * pow(((double) M_PI), 7.0)) - fma((pow(((double) M_PI), 2.0) * -0.5), (t_2 - t_4), fma(0.041666666666666664, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 4.0))), (pow(((double) M_PI), 7.0) * -0.001388888888888889))))) + (t_3 * (t_0 * t_5))) + (t_0 * (t_6 / ((double) M_PI)))))) + (pow(l, 3.0) * t_6)));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F_m) / F_m);
	}
	return tmp;
}

Julia

F_m = abs(F)
function code(F_m, l)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64((F_m ^ 2.0) / (pi ^ 2.0))
	t_2 = Float64(0.008333333333333333 * (pi ^ 5.0))
	t_3 = Float64((F_m ^ 2.0) / (pi ^ 3.0))
	t_4 = fma(-0.5, Float64((pi ^ 3.0) * Float64((pi ^ 2.0) * 0.3333333333333333)), Float64((pi ^ 5.0) * 0.041666666666666664))
	t_5 = Float64(t_4 - t_2)
	t_6 = Float64(Float64(t_3 * (t_0 ^ 2.0)) + Float64(t_1 * t_5))
	tmp = 0.0
	if (F_m <= 1.62e-161)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F_m) / F_m));
	elseif (F_m <= 5e-52)
		tmp = Float64(Float64(pi * l) - Float64(1.0 / Float64(Float64(Float64(Float64((F_m ^ 2.0) / Float64(pi * l)) - Float64(t_1 * Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)))) - Float64((l ^ 5.0) * Float64(Float64(Float64(t_1 * Float64(Float64(-0.0001984126984126984 * (pi ^ 7.0)) - fma(Float64((pi ^ 2.0) * -0.5), Float64(t_2 - t_4), fma(0.041666666666666664, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 4.0))), Float64((pi ^ 7.0) * -0.001388888888888889))))) + Float64(t_3 * Float64(t_0 * t_5))) + Float64(t_0 * Float64(t_6 / pi))))) + Float64((l ^ 3.0) * t_6))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F_m) / F_m));
	end
	return tmp
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F$95$m, 1.62e-161], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[F$95$m, 5e-52], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(N[(N[(N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[l, 5.0], $MachinePrecision] * N[(N[(N[(t$95$1 * N[(N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * N[(t$95$2 - t$95$4), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(t$95$6 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.62e-161

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

   if 1.62e-161 < F < 5e-52

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 91.3%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-1 \cdot \frac{{F}^{2} \cdot {\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}^{2}}{{\pi}^{3}} + \frac{{F}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)}{{\pi}^{2}}\right)\right) + \left(-1 \cdot \left({\ell}^{5} \cdot \left(-1 \cdot \frac{\left(-1 \cdot \frac{{F}^{2} \cdot {\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}^{2}}{{\pi}^{3}} + \frac{{F}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)}{{\pi}^{2}}\right) \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{\pi} + \left(-1 \cdot \frac{{F}^{2} \cdot \left(\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right) \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)\right)}{{\pi}^{3}} + \frac{{F}^{2} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)\right) + \left(-0.001388888888888889 \cdot {\pi}^{7} + 0.041666666666666664 \cdot \left({\pi}^{4} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)\right)\right)\right)}{{\pi}^{2}}\right)\right)\right) + \left(-1 \cdot \frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{{F}^{2}}{\ell \cdot \pi}\right)\right)}} \]

   6. Simplified 91.3%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(\left(\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)\right)\right) - {\ell}^{5} \cdot \left(\left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right), \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)\right) - \frac{\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}{\pi} \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)\right)\right) - {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}\right)}} \]

   if 5e-52 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \ell\right)}{F}}{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.62 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(\left(\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) - {\ell}^{5} \cdot \left(\left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right), \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) + \frac{{F}^{2}}{{\pi}^{3}} \cdot \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)\right) + \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \frac{\frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)}{\pi}\right)\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 86.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} t_0 := \frac{{F_m}^{2}}{{\pi}^{2}}\\ \mathbf{if}\;F_m \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F_m}}{F_m}\\ \mathbf{elif}\;F_m \leq 3 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\frac{{F_m}^{2}}{\pi \cdot \ell} - t_0 \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{{F_m}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} + t_0 \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F_m}}{F_m}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (let* ((t_0 (/ (pow F_m 2.0) (pow PI 2.0))))
   (if (<= F_m 1.1e-161)
     (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F_m) F_m))
     (if (<= F_m 3e-52)
       (+
        (* PI l)
        (/
         -1.0
         (+
          (-
           (/ (pow F_m 2.0) (* PI l))
           (* t_0 (* (pow PI 3.0) (* l 0.3333333333333333))))
          (*
           (pow l 3.0)
           (+
            (*
             (/ (pow F_m 2.0) (pow PI 3.0))
             (pow (* (pow PI 3.0) 0.3333333333333333) 2.0))
            (*
             t_0
             (-
              (fma
               -0.5
               (* (pow PI 3.0) (* (pow PI 2.0) 0.3333333333333333))
               (* (pow PI 5.0) 0.041666666666666664))
              (* 0.008333333333333333 (pow PI 5.0)))))))))
       (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F_m) F_m))))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double t_0 = pow(F_m, 2.0) / pow(((double) M_PI), 2.0);
	double tmp;
	if (F_m <= 1.1e-161) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F_m) / F_m);
	} else if (F_m <= 3e-52) {
		tmp = (((double) M_PI) * l) + (-1.0 / (((pow(F_m, 2.0) / (((double) M_PI) * l)) - (t_0 * (pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)))) + (pow(l, 3.0) * (((pow(F_m, 2.0) / pow(((double) M_PI), 3.0)) * pow((pow(((double) M_PI), 3.0) * 0.3333333333333333), 2.0)) + (t_0 * (fma(-0.5, (pow(((double) M_PI), 3.0) * (pow(((double) M_PI), 2.0) * 0.3333333333333333)), (pow(((double) M_PI), 5.0) * 0.041666666666666664)) - (0.008333333333333333 * pow(((double) M_PI), 5.0))))))));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F_m) / F_m);
	}
	return tmp;
}

Julia

F_m = abs(F)
function code(F_m, l)
	t_0 = Float64((F_m ^ 2.0) / (pi ^ 2.0))
	tmp = 0.0
	if (F_m <= 1.1e-161)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F_m) / F_m));
	elseif (F_m <= 3e-52)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64(Float64((F_m ^ 2.0) / Float64(pi * l)) - Float64(t_0 * Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)))) + Float64((l ^ 3.0) * Float64(Float64(Float64((F_m ^ 2.0) / (pi ^ 3.0)) * (Float64((pi ^ 3.0) * 0.3333333333333333) ^ 2.0)) + Float64(t_0 * Float64(fma(-0.5, Float64((pi ^ 3.0) * Float64((pi ^ 2.0) * 0.3333333333333333)), Float64((pi ^ 5.0) * 0.041666666666666664)) - Float64(0.008333333333333333 * (pi ^ 5.0)))))))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F_m) / F_m));
	end
	return tmp
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := Block[{t$95$0 = N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F$95$m, 1.1e-161], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[F$95$m, 3e-52], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(-0.5 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.10000000000000001e-161

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

   if 1.10000000000000001e-161 < F < 3e-52

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 86.8%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-1 \cdot \frac{{F}^{2} \cdot {\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}^{2}}{{\pi}^{3}} + \frac{{F}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)}{{\pi}^{2}}\right)\right) + \left(-1 \cdot \frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{{F}^{2}}{\ell \cdot \pi}\right)}} \]

   6. Simplified 86.8%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)\right)\right) - {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}\right)}} \]

   if 3e-52 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \ell\right)}{F}}{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left({\pi}^{2} \cdot 0.3333333333333333\right), {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} \mathbf{if}\;F_m \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F_m}}{F_m}\\ \mathbf{elif}\;F_m \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{{F_m}^{2}}{\pi \cdot \ell} - \frac{{F_m}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F_m}}{F_m}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (if (<= F_m 4.5e-171)
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F_m) F_m))
   (if (<= F_m 2e-51)
     (+
      (* PI l)
      (/
       -1.0
       (-
        (/ (pow F_m 2.0) (* PI l))
        (*
         (/ (pow F_m 2.0) (pow PI 2.0))
         (* (pow PI 3.0) (* l 0.3333333333333333))))))
     (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F_m) F_m)))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double tmp;
	if (F_m <= 4.5e-171) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F_m) / F_m);
	} else if (F_m <= 2e-51) {
		tmp = (((double) M_PI) * l) + (-1.0 / ((pow(F_m, 2.0) / (((double) M_PI) * l)) - ((pow(F_m, 2.0) / pow(((double) M_PI), 2.0)) * (pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)))));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F_m) / F_m);
	}
	return tmp;
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	double tmp;
	if (F_m <= 4.5e-171) {
		tmp = (Math.PI * l) - ((Math.tan((l * Math.pow(Math.sqrt(Math.PI), 2.0))) / F_m) / F_m);
	} else if (F_m <= 2e-51) {
		tmp = (Math.PI * l) + (-1.0 / ((Math.pow(F_m, 2.0) / (Math.PI * l)) - ((Math.pow(F_m, 2.0) / Math.pow(Math.PI, 2.0)) * (Math.pow(Math.PI, 3.0) * (l * 0.3333333333333333)))));
	} else {
		tmp = (Math.PI * l) - ((Math.tan((l * Math.cbrt(Math.pow(Math.PI, 3.0)))) / F_m) / F_m);
	}
	return tmp;
}

Julia

F_m = abs(F)
function code(F_m, l)
	tmp = 0.0
	if (F_m <= 4.5e-171)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F_m) / F_m));
	elseif (F_m <= 2e-51)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64((F_m ^ 2.0) / Float64(pi * l)) - Float64(Float64((F_m ^ 2.0) / (pi ^ 2.0)) * Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333))))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F_m) / F_m));
	end
	return tmp
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := If[LessEqual[F$95$m, 4.5e-171], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[F$95$m, 2e-51], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[F$95$m, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.5000000000000004e-171

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

   if 4.5000000000000004e-171 < F < 2e-51

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 77.9%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{-1 \cdot \frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{{F}^{2}}{\ell \cdot \pi}}} \]
   5. Step-by-step derivation

      1. +-commutative 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\ell \cdot \pi} + -1 \cdot \frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}}} \]

      2. mul-1-neg 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} + \color{blue}{\left(-\frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}\right)}} \]

      3. unsub-neg 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}}} \]

      4. associate-/l* 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \color{blue}{\frac{{F}^{2}}{\frac{{\pi}^{2}}{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}}}} \]

      5. associate-/r/ 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \color{blue}{\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}} \]

      6. *-commutative 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right) \cdot \ell\right)}} \]

      7. distribute-rgt-out-- 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\color{blue}{\left({\pi}^{3} \cdot \left(-0.16666666666666666 - -0.5\right)\right)} \cdot \ell\right)} \]

      8. metadata-eval 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\left({\pi}^{3} \cdot \color{blue}{0.3333333333333333}\right) \cdot \ell\right)} \]

      9. associate-*l* 77.9%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)\right)}} \]

   6. Simplified 77.9%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\ell \cdot \pi} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)\right)}} \]

   if 2e-51 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \ell\right)}{F}}{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \]

Alternative 4: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F_m}{\ell}}}{F_m}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F_m}}{F_m}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (if (<= (* PI l) 500000000.0)
   (- (* PI l) (/ (/ PI (/ F_m l)) F_m))
   (+ (* PI l) (/ (/ (tan (* PI l)) F_m) F_m))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 500000000.0) {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / (F_m / l)) / F_m);
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) / F_m) / F_m);
	}
	return tmp;
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	double tmp;
	if ((Math.PI * l) <= 500000000.0) {
		tmp = (Math.PI * l) - ((Math.PI / (F_m / l)) / F_m);
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) / F_m) / F_m);
	}
	return tmp;
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	tmp = 0
	if (math.pi * l) <= 500000000.0:
		tmp = (math.pi * l) - ((math.pi / (F_m / l)) / F_m)
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) / F_m) / F_m)
	return tmp

Julia

F_m = abs(F)
function code(F_m, l)
	tmp = 0.0
	if (Float64(pi * l) <= 500000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F_m / l)) / F_m));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) / F_m) / F_m));
	end
	return tmp
end

MATLAB

F_m = abs(F);
function tmp_2 = code(F_m, l)
	tmp = 0.0;
	if ((pi * l) <= 500000000.0)
		tmp = (pi * l) - ((pi / (F_m / l)) / F_m);
	else
		tmp = (pi * l) + ((tan((pi * l)) / F_m) / F_m);
	end
	tmp_2 = tmp;
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 500000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F$95$m / l), $MachinePrecision]), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 78.1%

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

      1. associate-*l/ 78.2%

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

      2. *-un-lft-identity 78.2%

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

      3. associate-/r* 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in l around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/l* 79.3%

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

   6. Simplified 79.3%

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

   if 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 66.5%

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

      1. fma-neg 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

      3. sqr-neg 66.5%

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

      4. distribute-neg-frac 66.5%

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

      5. metadata-eval 66.5%

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

      6. distribute-lft-neg-out 66.5%

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

      7. neg-mul-1 66.5%

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

      8. associate-/r* 66.5%

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

      9. metadata-eval 66.5%

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

      10. associate-*l/ 66.5%

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

      11. *-lft-identity 66.5%

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

      12. associate-/l/ 66.5%

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

   3. Simplified 66.5%

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

      1. fma-udef 66.5%

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

      2. +-commutative 66.5%

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

      3. associate-/l/ 66.5%

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

      4. associate-/r* 66.5%

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

      5. add-sqr-sqrt 33.4%

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

      6. sqrt-unprod 66.4%

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

      7. sqr-neg 66.4%

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

      8. sqrt-prod 32.9%

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

      9. add-sqr-sqrt 66.5%

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

      10. associate-/r* 66.5%

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

      11. div-inv 66.5%

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

      12. pow2 66.5%

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

      13. pow-flip 66.5%

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

      14. metadata-eval 66.5%

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

   5. Applied egg-rr 66.5%

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

      1. metadata-eval 66.5%

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

      2. pow-flip 66.5%

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

      3. pow2 66.5%

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

      4. div-inv 66.5%

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

      5. associate-/r* 66.5%

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

   7. Applied egg-rr 66.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 5: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F_m}{\ell}}}{F_m}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F_m \cdot F_m}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (if (<= (* PI l) 2e-117)
   (- (* PI l) (/ (/ PI (/ F_m l)) F_m))
   (- (* PI l) (/ (tan (* PI l)) (* F_m F_m)))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 2e-117) {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / (F_m / l)) / F_m);
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F_m * F_m));
	}
	return tmp;
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	double tmp;
	if ((Math.PI * l) <= 2e-117) {
		tmp = (Math.PI * l) - ((Math.PI / (F_m / l)) / F_m);
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F_m * F_m));
	}
	return tmp;
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	tmp = 0
	if (math.pi * l) <= 2e-117:
		tmp = (math.pi * l) - ((math.pi / (F_m / l)) / F_m)
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F_m * F_m))
	return tmp

Julia

F_m = abs(F)
function code(F_m, l)
	tmp = 0.0
	if (Float64(pi * l) <= 2e-117)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F_m / l)) / F_m));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F_m * F_m)));
	end
	return tmp
end

MATLAB

F_m = abs(F);
function tmp_2 = code(F_m, l)
	tmp = 0.0;
	if ((pi * l) <= 2e-117)
		tmp = (pi * l) - ((pi / (F_m / l)) / F_m);
	else
		tmp = (pi * l) - (tan((pi * l)) / (F_m * F_m));
	end
	tmp_2 = tmp;
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 2e-117], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F$95$m / l), $MachinePrecision]), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F$95$m * F$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2.00000000000000006e-117

   1. Initial program 75.0%

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

      1. associate-*l/ 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. associate-/r* 84.6%

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

   3. Applied egg-rr 84.6%

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

   4. Taylor expanded in l around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-/l* 76.3%

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

   6. Simplified 76.3%

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

   if 2.00000000000000006e-117 < (*.f64 (PI.f64) l)

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. associate-*l/ 74.9%

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

      3. sqr-neg 74.9%

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

      4. *-lft-identity 74.9%

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

   3. Simplified 74.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 6: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F_m}{\ell}}}{F_m}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot {F_m}^{-2}\\ \end{array} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (if (<= (* PI l) 500000000.0)
   (- (* PI l) (/ (/ PI (/ F_m l)) F_m))
   (+ (* PI l) (* (* PI l) (pow F_m -2.0)))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 500000000.0) {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / (F_m / l)) / F_m);
	} else {
		tmp = (((double) M_PI) * l) + ((((double) M_PI) * l) * pow(F_m, -2.0));
	}
	return tmp;
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	double tmp;
	if ((Math.PI * l) <= 500000000.0) {
		tmp = (Math.PI * l) - ((Math.PI / (F_m / l)) / F_m);
	} else {
		tmp = (Math.PI * l) + ((Math.PI * l) * Math.pow(F_m, -2.0));
	}
	return tmp;
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	tmp = 0
	if (math.pi * l) <= 500000000.0:
		tmp = (math.pi * l) - ((math.pi / (F_m / l)) / F_m)
	else:
		tmp = (math.pi * l) + ((math.pi * l) * math.pow(F_m, -2.0))
	return tmp

Julia

F_m = abs(F)
function code(F_m, l)
	tmp = 0.0
	if (Float64(pi * l) <= 500000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F_m / l)) / F_m));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(pi * l) * (F_m ^ -2.0)));
	end
	return tmp
end

MATLAB

F_m = abs(F);
function tmp_2 = code(F_m, l)
	tmp = 0.0;
	if ((pi * l) <= 500000000.0)
		tmp = (pi * l) - ((pi / (F_m / l)) / F_m);
	else
		tmp = (pi * l) + ((pi * l) * (F_m ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 500000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F$95$m / l), $MachinePrecision]), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(Pi * l), $MachinePrecision] * N[Power[F$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 78.1%

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

      1. associate-*l/ 78.2%

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

      2. *-un-lft-identity 78.2%

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

      3. associate-/r* 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in l around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/l* 79.3%

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

   6. Simplified 79.3%

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

   if 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 66.5%

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

      1. fma-neg 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

      3. sqr-neg 66.5%

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

      4. distribute-neg-frac 66.5%

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

      5. metadata-eval 66.5%

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

      6. distribute-lft-neg-out 66.5%

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

      7. neg-mul-1 66.5%

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

      8. associate-/r* 66.5%

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

      9. metadata-eval 66.5%

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

      10. associate-*l/ 66.5%

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

      11. *-lft-identity 66.5%

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

      12. associate-/l/ 66.5%

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

   3. Simplified 66.5%

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

      1. fma-udef 66.5%

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

      2. +-commutative 66.5%

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

      3. associate-/l/ 66.5%

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

      4. associate-/r* 66.5%

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

      5. add-sqr-sqrt 33.4%

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

      6. sqrt-unprod 66.4%

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

      7. sqr-neg 66.4%

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

      8. sqrt-prod 32.9%

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

      9. add-sqr-sqrt 66.5%

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

      10. associate-/r* 66.5%

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

      11. div-inv 66.5%

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

      12. pow2 66.5%

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

      13. pow-flip 66.5%

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

      14. metadata-eval 66.5%

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

   5. Applied egg-rr 66.5%

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

   6. Taylor expanded in l around 0 57.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot {F}^{-2}\\ \end{array} \]

Alternative 7: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \pi \cdot \ell + \frac{-1}{F_m \cdot \frac{F_m}{\tan \left(\pi \cdot \ell\right)}} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (+ (* PI l) (/ -1.0 (* F_m (/ F_m (tan (* PI l)))))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	return (((double) M_PI) * l) + (-1.0 / (F_m * (F_m / tan((((double) M_PI) * l)))));
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	return (Math.PI * l) + (-1.0 / (F_m * (F_m / Math.tan((Math.PI * l)))));
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	return (math.pi * l) + (-1.0 / (F_m * (F_m / math.tan((math.pi * l)))))

Julia

F_m = abs(F)
function code(F_m, l)
	return Float64(Float64(pi * l) + Float64(-1.0 / Float64(F_m * Float64(F_m / tan(Float64(pi * l))))))
end

MATLAB

F_m = abs(F);
function tmp = code(F_m, l)
	tmp = (pi * l) + (-1.0 / (F_m * (F_m / tan((pi * l)))));
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F$95$m * N[(F$95$m / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-/r/ 75.0%

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

   2. associate-/l* 81.1%

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

   3. clear-num 81.1%

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

   4. add-sqr-sqrt 39.5%

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

   5. sqrt-prod 65.1%

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

   6. sqr-neg 65.1%

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

   7. sqrt-unprod 29.2%

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

   8. add-sqr-sqrt 56.7%

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

   9. associate-/r* 56.7%

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

   10. clear-num 56.7%

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

   11. add-sqr-sqrt 29.2%

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

   12. sqrt-unprod 65.1%

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

   13. sqr-neg 65.1%

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

   14. sqrt-prod 35.9%

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

   15. add-sqr-sqrt 75.0%

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

   16. pow2 75.0%

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

3. Applied egg-rr 75.0%

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

   1. div-inv 75.0%

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

   2. unpow2 75.0%

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

   3. associate-*l* 81.1%

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

   4. div-inv 81.1%

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

   5. *-commutative 81.1%

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

5. Applied egg-rr 81.1%

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

6. Final simplification 81.1%

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

Alternative 8: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F_m}}{F_m} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l)
 :precision binary64
 (- (* PI l) (/ (/ (tan (* PI l)) F_m) F_m)))

C

F_m = fabs(F);
double code(double F_m, double l) {
	return (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F_m) / F_m);
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	return (Math.PI * l) - ((Math.tan((Math.PI * l)) / F_m) / F_m);
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	return (math.pi * l) - ((math.tan((math.pi * l)) / F_m) / F_m)

Julia

F_m = abs(F)
function code(F_m, l)
	return Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F_m) / F_m))
end

MATLAB

F_m = abs(F);
function tmp = code(F_m, l)
	tmp = (pi * l) - ((tan((pi * l)) / F_m) / F_m);
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F$95$m), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-*l/ 75.0%

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

   2. *-un-lft-identity 75.0%

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

   3. associate-/r* 81.1%

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

3. Applied egg-rr 81.1%

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

4. Final simplification 81.1%

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

Alternative 9: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \pi \cdot \ell - \frac{\pi}{F_m} \cdot \frac{\ell}{F_m} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l) :precision binary64 (- (* PI l) (* (/ PI F_m) (/ l F_m))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	return (((double) M_PI) * l) - ((((double) M_PI) / F_m) * (l / F_m));
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	return (Math.PI * l) - ((Math.PI / F_m) * (l / F_m));
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	return (math.pi * l) - ((math.pi / F_m) * (l / F_m))

Julia

F_m = abs(F)
function code(F_m, l)
	return Float64(Float64(pi * l) - Float64(Float64(pi / F_m) * Float64(l / F_m)))
end

MATLAB

F_m = abs(F);
function tmp = code(F_m, l)
	tmp = (pi * l) - ((pi / F_m) * (l / F_m));
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F$95$m), $MachinePrecision] * N[(l / F$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-*l/ 75.0%

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

   3. sqr-neg 75.0%

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

   4. *-lft-identity 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. times-frac 72.4%

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

6. Applied egg-rr 72.4%

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

7. Final simplification 72.4%

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

Alternative 10: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \pi \cdot \ell - \frac{\pi}{F_m \cdot \frac{F_m}{\ell}} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l) :precision binary64 (- (* PI l) (/ PI (* F_m (/ F_m l)))))

C

F_m = fabs(F);
double code(double F_m, double l) {
	return (((double) M_PI) * l) - (((double) M_PI) / (F_m * (F_m / l)));
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	return (Math.PI * l) - (Math.PI / (F_m * (F_m / l)));
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	return (math.pi * l) - (math.pi / (F_m * (F_m / l)))

Julia

F_m = abs(F)
function code(F_m, l)
	return Float64(Float64(pi * l) - Float64(pi / Float64(F_m * Float64(F_m / l))))
end

MATLAB

F_m = abs(F);
function tmp = code(F_m, l)
	tmp = (pi * l) - (pi / (F_m * (F_m / l)));
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F$95$m * N[(F$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-*l/ 75.0%

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

   3. sqr-neg 75.0%

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

   4. *-lft-identity 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. times-frac 72.4%

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

6. Applied egg-rr 72.4%

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

   1. *-commutative 72.4%

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

   2. clear-num 72.4%

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

   3. frac-times 72.4%

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

   4. *-un-lft-identity 72.4%

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

8. Applied egg-rr 72.4%

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

9. Final simplification 72.4%

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

Alternative 11: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} F_m = \left|F\right| \\ \pi \cdot \ell - \frac{\frac{\pi}{\frac{F_m}{\ell}}}{F_m} \end{array} \]

FPCore

F_m = (fabs.f64 F)
(FPCore (F_m l) :precision binary64 (- (* PI l) (/ (/ PI (/ F_m l)) F_m)))

C

F_m = fabs(F);
double code(double F_m, double l) {
	return (((double) M_PI) * l) - ((((double) M_PI) / (F_m / l)) / F_m);
}

Java

F_m = Math.abs(F);
public static double code(double F_m, double l) {
	return (Math.PI * l) - ((Math.PI / (F_m / l)) / F_m);
}

Python

F_m = math.fabs(F)
def code(F_m, l):
	return (math.pi * l) - ((math.pi / (F_m / l)) / F_m)

Julia

F_m = abs(F)
function code(F_m, l)
	return Float64(Float64(pi * l) - Float64(Float64(pi / Float64(F_m / l)) / F_m))
end

MATLAB

F_m = abs(F);
function tmp = code(F_m, l)
	tmp = (pi * l) - ((pi / (F_m / l)) / F_m);
end

Wolfram

F_m = N[Abs[F], $MachinePrecision]
code[F$95$m_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / N[(F$95$m / l), $MachinePrecision]), $MachinePrecision] / F$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-*l/ 75.0%

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

   2. *-un-lft-identity 75.0%

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

   3. associate-/r* 81.1%

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

3. Applied egg-rr 81.1%

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

4. Taylor expanded in l around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. associate-/l* 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))