VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.0% → 86.5%

Time: 38.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. F = -1.6053342310434395e-298
  2. l = 3114781257601063.5

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: 76.0% 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: 86.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, t_0 \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_2 := \mathsf{fma}\left(-1, \frac{{F}^{2}}{\frac{{\pi}^{3}}{{t_0}^{2}}}, \frac{{F}^{2}}{\frac{{\pi}^{2}}{t_1}}\right)\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot l_m + \frac{-1}{\mathsf{fma}\left(-1, {l_m}^{3} \cdot t_2, \mathsf{fma}\left(-1, {l_m}^{5} \cdot \mathsf{fma}\left(-1, \frac{t_2}{\frac{\pi}{t_0}}, \mathsf{fma}\left(-1, \frac{{F}^{2}}{\frac{{\pi}^{3}}{t_0 \cdot t_1}}, \frac{{F}^{2}}{\frac{{\pi}^{2}}{-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(-0.5, {\pi}^{2} \cdot t_1, \mathsf{fma}\left(-0.001388888888888889, {\pi}^{7}, 0.041666666666666664 \cdot \left(t_0 \cdot {\pi}^{4}\right)\right)\right)}}\right)\right), \mathsf{fma}\left(-1, \frac{t_0 \cdot \left(l_m \cdot {F}^{2}\right)}{{\pi}^{2}}, \frac{\frac{{F}^{2}}{\pi}}{l_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1
         (-
          (* 0.008333333333333333 (pow PI 5.0))
          (fma
           -0.5
           (* t_0 (pow PI 2.0))
           (* (pow PI 5.0) 0.041666666666666664))))
        (t_2
         (fma
          -1.0
          (/ (pow F 2.0) (/ (pow PI 3.0) (pow t_0 2.0)))
          (/ (pow F 2.0) (/ (pow PI 2.0) t_1)))))
   (*
    l_s
    (if (<= (* F F) 0.0)
      (- (* PI l_m) (/ (/ (tan (expm1 (log1p (* PI l_m)))) F) F))
      (if (<= (* F F) 1e-73)
        (+
         (* PI l_m)
         (/
          -1.0
          (fma
           -1.0
           (* (pow l_m 3.0) t_2)
           (fma
            -1.0
            (*
             (pow l_m 5.0)
             (fma
              -1.0
              (/ t_2 (/ PI t_0))
              (fma
               -1.0
               (/ (pow F 2.0) (/ (pow PI 3.0) (* t_0 t_1)))
               (/
                (pow F 2.0)
                (/
                 (pow PI 2.0)
                 (-
                  (* -0.0001984126984126984 (pow PI 7.0))
                  (fma
                   -0.5
                   (* (pow PI 2.0) t_1)
                   (fma
                    -0.001388888888888889
                    (pow PI 7.0)
                    (* 0.041666666666666664 (* t_0 (pow PI 4.0)))))))))))
            (fma
             -1.0
             (/ (* t_0 (* l_m (pow F 2.0))) (pow PI 2.0))
             (/ (/ (pow F 2.0) PI) l_m))))))
        (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = (0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, (t_0 * pow(((double) M_PI), 2.0)), (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_2 = fma(-1.0, (pow(F, 2.0) / (pow(((double) M_PI), 3.0) / pow(t_0, 2.0))), (pow(F, 2.0) / (pow(((double) M_PI), 2.0) / t_1)));
	double tmp;
	if ((F * F) <= 0.0) {
		tmp = (((double) M_PI) * l_m) - ((tan(expm1(log1p((((double) M_PI) * l_m)))) / F) / F);
	} else if ((F * F) <= 1e-73) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / fma(-1.0, (pow(l_m, 3.0) * t_2), fma(-1.0, (pow(l_m, 5.0) * fma(-1.0, (t_2 / (((double) M_PI) / t_0)), fma(-1.0, (pow(F, 2.0) / (pow(((double) M_PI), 3.0) / (t_0 * t_1))), (pow(F, 2.0) / (pow(((double) M_PI), 2.0) / ((-0.0001984126984126984 * pow(((double) M_PI), 7.0)) - fma(-0.5, (pow(((double) M_PI), 2.0) * t_1), fma(-0.001388888888888889, pow(((double) M_PI), 7.0), (0.041666666666666664 * (t_0 * pow(((double) M_PI), 4.0))))))))))), fma(-1.0, ((t_0 * (l_m * pow(F, 2.0))) / pow(((double) M_PI), 2.0)), ((pow(F, 2.0) / ((double) M_PI)) / l_m)))));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64(t_0 * (pi ^ 2.0)), Float64((pi ^ 5.0) * 0.041666666666666664)))
	t_2 = fma(-1.0, Float64((F ^ 2.0) / Float64((pi ^ 3.0) / (t_0 ^ 2.0))), Float64((F ^ 2.0) / Float64((pi ^ 2.0) / t_1)))
	tmp = 0.0
	if (Float64(F * F) <= 0.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(expm1(log1p(Float64(pi * l_m)))) / F) / F));
	elseif (Float64(F * F) <= 1e-73)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / fma(-1.0, Float64((l_m ^ 3.0) * t_2), fma(-1.0, Float64((l_m ^ 5.0) * fma(-1.0, Float64(t_2 / Float64(pi / t_0)), fma(-1.0, Float64((F ^ 2.0) / Float64((pi ^ 3.0) / Float64(t_0 * t_1))), Float64((F ^ 2.0) / Float64((pi ^ 2.0) / Float64(Float64(-0.0001984126984126984 * (pi ^ 7.0)) - fma(-0.5, Float64((pi ^ 2.0) * t_1), fma(-0.001388888888888889, (pi ^ 7.0), Float64(0.041666666666666664 * Float64(t_0 * (pi ^ 4.0))))))))))), fma(-1.0, Float64(Float64(t_0 * Float64(l_m * (F ^ 2.0))) / (pi ^ 2.0)), Float64(Float64((F ^ 2.0) / pi) / l_m))))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(t$95$0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(N[Power[F, 2.0], $MachinePrecision] / N[(N[Power[Pi, 3.0], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(F * F), $MachinePrecision], 0.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-73], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(-1.0 * N[(N[Power[l$95$m, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(-1.0 * N[(N[Power[l$95$m, 5.0], $MachinePrecision] * N[(-1.0 * N[(t$95$2 / N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[Power[F, 2.0], $MachinePrecision] / N[(N[Power[Pi, 3.0], $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] / N[(N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(-0.001388888888888889 * N[Power[Pi, 7.0], $MachinePrecision] + N[(0.041666666666666664 * N[(t$95$0 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(t$95$0 * N[(l$95$m * N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[F, 2.0], $MachinePrecision] / Pi), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 0.0

   1. Initial program 30.0%

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

      1. associate-*l/ 30.0%

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

      2. *-un-lft-identity 30.0%

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

      3. associate-/r* 51.2%

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

   4. Applied egg-rr 51.2%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   if 0.0 < (*.f64 F F) < 9.99999999999999997e-74

   1. Initial program 69.3%

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

      1. associate-/r/ 69.2%

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

      2. associate-/l* 69.2%

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

      3. clear-num 69.3%

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

      4. add-sqr-sqrt 36.0%

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

      5. sqrt-prod 48.4%

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

      6. sqr-neg 48.4%

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

      7. sqrt-unprod 12.3%

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

      8. add-sqr-sqrt 20.3%

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

      9. associate-/l/ 20.3%

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

      10. clear-num 20.3%

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

      11. add-sqr-sqrt 12.3%

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

      12. sqrt-unprod 48.3%

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

      13. sqr-neg 48.3%

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

      14. sqrt-prod 36.0%

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

      15. add-sqr-sqrt 69.2%

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

      16. pow2 69.2%

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

   4. Applied egg-rr 69.2%

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

   5. Taylor expanded in l around 0 93.3%

      \[\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)}} \]

   7. Simplified 93.4%

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

   if 9.99999999999999997e-74 < (*.f64 F F)

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\mathsf{fma}\left(-1, {\ell}^{3} \cdot \mathsf{fma}\left(-1, \frac{{F}^{2}}{\frac{{\pi}^{3}}{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}}, \frac{{F}^{2}}{\frac{{\pi}^{2}}{0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)}}\right), \mathsf{fma}\left(-1, {\ell}^{5} \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{F}^{2}}{\frac{{\pi}^{3}}{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}}, \frac{{F}^{2}}{\frac{{\pi}^{2}}{0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)}}\right)}{\frac{\pi}{{\pi}^{3} \cdot 0.3333333333333333}}, \mathsf{fma}\left(-1, \frac{{F}^{2}}{\frac{{\pi}^{3}}{\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\right)}}, \frac{{F}^{2}}{\frac{{\pi}^{2}}{-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(-0.5, {\pi}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\right), \mathsf{fma}\left(-0.001388888888888889, {\pi}^{7}, 0.041666666666666664 \cdot \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{4}\right)\right)\right)}}\right)\right), \mathsf{fma}\left(-1, \frac{\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(\ell \cdot {F}^{2}\right)}{{\pi}^{2}}, \frac{\frac{{F}^{2}}{\pi}}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \mathsf{fma}\left(0.008333333333333333, {\pi}^{5}, \left(t_0 \cdot {\pi}^{2}\right) \cdot 0.5\right) + {\pi}^{5} \cdot -0.041666666666666664\\ t_2 := \frac{{F}^{2}}{{\pi}^{2}}\\ t_3 := \mathsf{fma}\left(t_2, t_1, \frac{-{F}^{2}}{\frac{{\pi}^{3}}{{t_0}^{2}}}\right)\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot l_m + \frac{-1}{\mathsf{fma}\left({l_m}^{5}, -\mathsf{fma}\left(t_2, -0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(t_0, 0.041666666666666664 \cdot {\pi}^{4}, \mathsf{fma}\left({\pi}^{2}, -0.5 \cdot t_1, {\pi}^{7} \cdot -0.001388888888888889\right)\right), -\mathsf{fma}\left(\frac{t_3}{\pi}, t_0, \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot t_1\right)\right)\right)\right), \mathsf{fma}\left(-1, \mathsf{fma}\left({l_m}^{3}, t_3, t_2 \cdot \left(l_m \cdot t_0\right)\right), \frac{{F}^{2}}{\pi \cdot l_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1
         (+
          (fma 0.008333333333333333 (pow PI 5.0) (* (* t_0 (pow PI 2.0)) 0.5))
          (* (pow PI 5.0) -0.041666666666666664)))
        (t_2 (/ (pow F 2.0) (pow PI 2.0)))
        (t_3 (fma t_2 t_1 (/ (- (pow F 2.0)) (/ (pow PI 3.0) (pow t_0 2.0))))))
   (*
    l_s
    (if (<= (* F F) 0.0)
      (- (* PI l_m) (/ (/ (tan (expm1 (log1p (* PI l_m)))) F) F))
      (if (<= (* F F) 1e-73)
        (+
         (* PI l_m)
         (/
          -1.0
          (fma
           (pow l_m 5.0)
           (-
            (fma
             t_2
             (-
              (* -0.0001984126984126984 (pow PI 7.0))
              (fma
               t_0
               (* 0.041666666666666664 (pow PI 4.0))
               (fma
                (pow PI 2.0)
                (* -0.5 t_1)
                (* (pow PI 7.0) -0.001388888888888889))))
             (-
              (fma
               (/ t_3 PI)
               t_0
               (*
                (/ (pow F 2.0) (pow PI 3.0))
                (* (pow PI 3.0) (* 0.3333333333333333 t_1)))))))
           (fma
            -1.0
            (fma (pow l_m 3.0) t_3 (* t_2 (* l_m t_0)))
            (/ (pow F 2.0) (* PI l_m))))))
        (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = fma(0.008333333333333333, pow(((double) M_PI), 5.0), ((t_0 * pow(((double) M_PI), 2.0)) * 0.5)) + (pow(((double) M_PI), 5.0) * -0.041666666666666664);
	double t_2 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double t_3 = fma(t_2, t_1, (-pow(F, 2.0) / (pow(((double) M_PI), 3.0) / pow(t_0, 2.0))));
	double tmp;
	if ((F * F) <= 0.0) {
		tmp = (((double) M_PI) * l_m) - ((tan(expm1(log1p((((double) M_PI) * l_m)))) / F) / F);
	} else if ((F * F) <= 1e-73) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / fma(pow(l_m, 5.0), -fma(t_2, ((-0.0001984126984126984 * pow(((double) M_PI), 7.0)) - fma(t_0, (0.041666666666666664 * pow(((double) M_PI), 4.0)), fma(pow(((double) M_PI), 2.0), (-0.5 * t_1), (pow(((double) M_PI), 7.0) * -0.001388888888888889)))), -fma((t_3 / ((double) M_PI)), t_0, ((pow(F, 2.0) / pow(((double) M_PI), 3.0)) * (pow(((double) M_PI), 3.0) * (0.3333333333333333 * t_1))))), fma(-1.0, fma(pow(l_m, 3.0), t_3, (t_2 * (l_m * t_0))), (pow(F, 2.0) / (((double) M_PI) * l_m)))));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64(fma(0.008333333333333333, (pi ^ 5.0), Float64(Float64(t_0 * (pi ^ 2.0)) * 0.5)) + Float64((pi ^ 5.0) * -0.041666666666666664))
	t_2 = Float64((F ^ 2.0) / (pi ^ 2.0))
	t_3 = fma(t_2, t_1, Float64(Float64(-(F ^ 2.0)) / Float64((pi ^ 3.0) / (t_0 ^ 2.0))))
	tmp = 0.0
	if (Float64(F * F) <= 0.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(expm1(log1p(Float64(pi * l_m)))) / F) / F));
	elseif (Float64(F * F) <= 1e-73)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / fma((l_m ^ 5.0), Float64(-fma(t_2, Float64(Float64(-0.0001984126984126984 * (pi ^ 7.0)) - fma(t_0, Float64(0.041666666666666664 * (pi ^ 4.0)), fma((pi ^ 2.0), Float64(-0.5 * t_1), Float64((pi ^ 7.0) * -0.001388888888888889)))), Float64(-fma(Float64(t_3 / pi), t_0, Float64(Float64((F ^ 2.0) / (pi ^ 3.0)) * Float64((pi ^ 3.0) * Float64(0.3333333333333333 * t_1))))))), fma(-1.0, fma((l_m ^ 3.0), t_3, Float64(t_2 * Float64(l_m * t_0))), Float64((F ^ 2.0) / Float64(pi * l_m))))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision] + N[(N[(t$95$0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1 + N[((-N[Power[F, 2.0], $MachinePrecision]) / N[(N[Power[Pi, 3.0], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(F * F), $MachinePrecision], 0.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-73], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(N[Power[l$95$m, 5.0], $MachinePrecision] * (-N[(t$95$2 * N[(N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(-0.5 * t$95$1), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(t$95$3 / Pi), $MachinePrecision] * t$95$0 + N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]) + N[(-1.0 * N[(N[Power[l$95$m, 3.0], $MachinePrecision] * t$95$3 + N[(t$95$2 * N[(l$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 0.0

   1. Initial program 30.0%

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

      1. associate-*l/ 30.0%

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

      2. *-un-lft-identity 30.0%

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

      3. associate-/r* 51.2%

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

   4. Applied egg-rr 51.2%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   if 0.0 < (*.f64 F F) < 9.99999999999999997e-74

   1. Initial program 69.3%

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

      1. associate-/r/ 69.2%

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

      2. associate-/l* 69.2%

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

      3. clear-num 69.3%

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

      4. add-sqr-sqrt 36.0%

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

      5. sqrt-prod 48.4%

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

      6. sqr-neg 48.4%

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

      7. sqrt-unprod 12.3%

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

      8. add-sqr-sqrt 20.3%

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

      9. associate-/l/ 20.3%

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

      10. clear-num 20.3%

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

      11. add-sqr-sqrt 12.3%

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

      12. sqrt-unprod 48.3%

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

      13. sqr-neg 48.3%

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

      14. sqrt-prod 36.0%

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

      15. add-sqr-sqrt 69.2%

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

      16. pow2 69.2%

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

   4. Applied egg-rr 69.2%

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

      1. add-log-exp 22.1%

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

   6. Applied egg-rr 22.1%

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

   7. Taylor expanded in l around 0 93.3%

      \[\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)}} \]

   9. Simplified 93.3%

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

   if 9.99999999999999997e-74 < (*.f64 F F)

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\mathsf{fma}\left({\ell}^{5}, -\mathsf{fma}\left(\frac{{F}^{2}}{{\pi}^{2}}, -0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left({\pi}^{3} \cdot 0.3333333333333333, 0.041666666666666664 \cdot {\pi}^{4}, \mathsf{fma}\left({\pi}^{2}, -0.5 \cdot \left(\mathsf{fma}\left(0.008333333333333333, {\pi}^{5}, \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}\right) \cdot 0.5\right) + {\pi}^{5} \cdot -0.041666666666666664\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right), -\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{F}^{2}}{{\pi}^{2}}, \mathsf{fma}\left(0.008333333333333333, {\pi}^{5}, \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}\right) \cdot 0.5\right) + {\pi}^{5} \cdot -0.041666666666666664, \frac{-{F}^{2}}{\frac{{\pi}^{3}}{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}}\right)}{\pi}, {\pi}^{3} \cdot 0.3333333333333333, \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.008333333333333333, {\pi}^{5}, \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}\right) \cdot 0.5\right) + {\pi}^{5} \cdot -0.041666666666666664\right)\right)\right)\right)\right), \mathsf{fma}\left(-1, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left(\frac{{F}^{2}}{{\pi}^{2}}, \mathsf{fma}\left(0.008333333333333333, {\pi}^{5}, \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}\right) \cdot 0.5\right) + {\pi}^{5} \cdot -0.041666666666666664, \frac{-{F}^{2}}{\frac{{\pi}^{3}}{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}}\right), \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)\right)\right), \frac{{F}^{2}}{\pi \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := 0.008333333333333333 \cdot {\pi}^{5}\\ t_1 := {\pi}^{2} \cdot -0.5\\ t_2 := \frac{{F}^{2}}{{\pi}^{3}}\\ t_3 := {\pi}^{3} \cdot 0.3333333333333333\\ t_4 := \mathsf{fma}\left(t_1, t_3, {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_5 := {t_3}^{2} \cdot t_2\\ t_6 := \frac{{F}^{2}}{{\pi}^{2}}\\ t_7 := t_0 - t_4\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot l_m + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot l_m} - t_6 \cdot \left({\pi}^{3} \cdot \left(l_m \cdot 0.3333333333333333\right)\right)\right) + {l_m}^{5} \cdot \left(t_3 \cdot \frac{t_6 \cdot t_7 - t_5}{\pi} + \left(t_2 \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot t_7\right)\right) + t_6 \cdot \left(\mathsf{fma}\left(t_1, t_7, \mathsf{fma}\left(0.041666666666666664, t_3 \cdot {\pi}^{4}, {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right)\right) + {l_m}^{3} \cdot \left(t_6 \cdot \left(t_4 - t_0\right) + t_5\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* 0.008333333333333333 (pow PI 5.0)))
        (t_1 (* (pow PI 2.0) -0.5))
        (t_2 (/ (pow F 2.0) (pow PI 3.0)))
        (t_3 (* (pow PI 3.0) 0.3333333333333333))
        (t_4 (fma t_1 t_3 (* (pow PI 5.0) 0.041666666666666664)))
        (t_5 (* (pow t_3 2.0) t_2))
        (t_6 (/ (pow F 2.0) (pow PI 2.0)))
        (t_7 (- t_0 t_4)))
   (*
    l_s
    (if (<= (* F F) 0.0)
      (- (* PI l_m) (/ (/ (tan (expm1 (log1p (* PI l_m)))) F) F))
      (if (<= (* F F) 1e-73)
        (+
         (* PI l_m)
         (/
          -1.0
          (+
           (+
            (-
             (/ (pow F 2.0) (* PI l_m))
             (* t_6 (* (pow PI 3.0) (* l_m 0.3333333333333333))))
            (*
             (pow l_m 5.0)
             (+
              (* t_3 (/ (- (* t_6 t_7) t_5) PI))
              (+
               (* t_2 (* (pow PI 3.0) (* 0.3333333333333333 t_7)))
               (*
                t_6
                (-
                 (fma
                  t_1
                  t_7
                  (fma
                   0.041666666666666664
                   (* t_3 (pow PI 4.0))
                   (* (pow PI 7.0) -0.001388888888888889)))
                 (* -0.0001984126984126984 (pow PI 7.0))))))))
           (* (pow l_m 3.0) (+ (* t_6 (- t_4 t_0)) t_5)))))
        (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = 0.008333333333333333 * pow(((double) M_PI), 5.0);
	double t_1 = pow(((double) M_PI), 2.0) * -0.5;
	double t_2 = pow(F, 2.0) / pow(((double) M_PI), 3.0);
	double t_3 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_4 = fma(t_1, t_3, (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_5 = pow(t_3, 2.0) * t_2;
	double t_6 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double t_7 = t_0 - t_4;
	double tmp;
	if ((F * F) <= 0.0) {
		tmp = (((double) M_PI) * l_m) - ((tan(expm1(log1p((((double) M_PI) * l_m)))) / F) / F);
	} else if ((F * F) <= 1e-73) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / ((((pow(F, 2.0) / (((double) M_PI) * l_m)) - (t_6 * (pow(((double) M_PI), 3.0) * (l_m * 0.3333333333333333)))) + (pow(l_m, 5.0) * ((t_3 * (((t_6 * t_7) - t_5) / ((double) M_PI))) + ((t_2 * (pow(((double) M_PI), 3.0) * (0.3333333333333333 * t_7))) + (t_6 * (fma(t_1, t_7, fma(0.041666666666666664, (t_3 * pow(((double) M_PI), 4.0)), (pow(((double) M_PI), 7.0) * -0.001388888888888889))) - (-0.0001984126984126984 * pow(((double) M_PI), 7.0)))))))) + (pow(l_m, 3.0) * ((t_6 * (t_4 - t_0)) + t_5))));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(0.008333333333333333 * (pi ^ 5.0))
	t_1 = Float64((pi ^ 2.0) * -0.5)
	t_2 = Float64((F ^ 2.0) / (pi ^ 3.0))
	t_3 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_4 = fma(t_1, t_3, Float64((pi ^ 5.0) * 0.041666666666666664))
	t_5 = Float64((t_3 ^ 2.0) * t_2)
	t_6 = Float64((F ^ 2.0) / (pi ^ 2.0))
	t_7 = Float64(t_0 - t_4)
	tmp = 0.0
	if (Float64(F * F) <= 0.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(expm1(log1p(Float64(pi * l_m)))) / F) / F));
	elseif (Float64(F * F) <= 1e-73)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(Float64(Float64(Float64((F ^ 2.0) / Float64(pi * l_m)) - Float64(t_6 * Float64((pi ^ 3.0) * Float64(l_m * 0.3333333333333333)))) + Float64((l_m ^ 5.0) * Float64(Float64(t_3 * Float64(Float64(Float64(t_6 * t_7) - t_5) / pi)) + Float64(Float64(t_2 * Float64((pi ^ 3.0) * Float64(0.3333333333333333 * t_7))) + Float64(t_6 * Float64(fma(t_1, t_7, fma(0.041666666666666664, Float64(t_3 * (pi ^ 4.0)), Float64((pi ^ 7.0) * -0.001388888888888889))) - Float64(-0.0001984126984126984 * (pi ^ 7.0)))))))) + Float64((l_m ^ 3.0) * Float64(Float64(t_6 * Float64(t_4 - t_0)) + t_5)))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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_] := Block[{t$95$0 = N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$3 + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[t$95$3, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 - t$95$4), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(F * F), $MachinePrecision], 0.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 1e-73], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 5.0], $MachinePrecision] * N[(N[(t$95$3 * N[(N[(N[(t$95$6 * t$95$7), $MachinePrecision] - t$95$5), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * N[(N[(t$95$1 * t$95$7 + N[(0.041666666666666664 * N[(t$95$3 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 3.0], $MachinePrecision] * N[(N[(t$95$6 * N[(t$95$4 - t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 F F) < 0.0

   1. Initial program 30.0%

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

      1. associate-*l/ 30.0%

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

      2. *-un-lft-identity 30.0%

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

      3. associate-/r* 51.2%

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

   4. Applied egg-rr 51.2%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   if 0.0 < (*.f64 F F) < 9.99999999999999997e-74

   1. Initial program 69.3%

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

      1. associate-/r/ 69.2%

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

      2. associate-/l* 69.2%

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

      3. clear-num 69.3%

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

      4. add-sqr-sqrt 36.0%

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

      5. sqrt-prod 48.4%

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

      6. sqr-neg 48.4%

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

      7. sqrt-unprod 12.3%

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

      8. add-sqr-sqrt 20.3%

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

      9. associate-/l/ 20.3%

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

      10. clear-num 20.3%

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

      11. add-sqr-sqrt 12.3%

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

      12. sqrt-unprod 48.3%

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

      13. sqr-neg 48.3%

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

      14. sqrt-prod 36.0%

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

      15. add-sqr-sqrt 69.2%

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

      16. pow2 69.2%

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

   4. Applied egg-rr 69.2%

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

   5. Taylor expanded in l around 0 93.3%

      \[\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)}} \]

   7. Simplified 93.3%

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

   if 9.99999999999999997e-74 < (*.f64 F F)

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

      4. sqr-neg 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 0:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \cdot F \leq 10^{-73}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + {\ell}^{5} \cdot \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \frac{\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) - {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}}}{\pi} + \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right), \mathsf{fma}\left(0.041666666666666664, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{4}, {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right) + {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \frac{{F}^{2}}{{\pi}^{3}}\\ t_2 := {t_0}^{2} \cdot t_1\\ t_3 := 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, t_0 \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_4 := \frac{{F}^{2}}{{\pi}^{2}}\\ t_5 := t_3 \cdot t_4\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 5.5 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-36}:\\ \;\;\;\;\pi \cdot l_m + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot l_m} - t_4 \cdot \left(l_m \cdot t_0\right)\right) + {l_m}^{5} \cdot \left(\frac{t_5 - t_2}{\frac{\pi}{t_0}} + \left(t_1 \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot t_3\right)\right) + t_4 \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, t_3, \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right)\right) + {l_m}^{3} \cdot \left(t_2 - t_5\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1 (/ (pow F 2.0) (pow PI 3.0)))
        (t_2 (* (pow t_0 2.0) t_1))
        (t_3
         (-
          (* 0.008333333333333333 (pow PI 5.0))
          (fma
           -0.5
           (* t_0 (pow PI 2.0))
           (* (pow PI 5.0) 0.041666666666666664))))
        (t_4 (/ (pow F 2.0) (pow PI 2.0)))
        (t_5 (* t_3 t_4)))
   (*
    l_s
    (if (<= F 5.5e-162)
      (- (* PI l_m) (/ (/ (tan (expm1 (log1p (* PI l_m)))) F) F))
      (if (<= F 1.48e-36)
        (+
         (* PI l_m)
         (/
          -1.0
          (+
           (+
            (- (/ (pow F 2.0) (* PI l_m)) (* t_4 (* l_m t_0)))
            (*
             (pow l_m 5.0)
             (+
              (/ (- t_5 t_2) (/ PI t_0))
              (+
               (* t_1 (* (pow PI 3.0) (* 0.3333333333333333 t_3)))
               (*
                t_4
                (-
                 (fma
                  (* (pow PI 2.0) -0.5)
                  t_3
                  (fma
                   0.041666666666666664
                   (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 4.0)))
                   (* (pow PI 7.0) -0.001388888888888889)))
                 (* -0.0001984126984126984 (pow PI 7.0))))))))
           (* (pow l_m 3.0) (- t_2 t_5)))))
        (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = pow(F, 2.0) / pow(((double) M_PI), 3.0);
	double t_2 = pow(t_0, 2.0) * t_1;
	double t_3 = (0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, (t_0 * pow(((double) M_PI), 2.0)), (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_4 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double t_5 = t_3 * t_4;
	double tmp;
	if (F <= 5.5e-162) {
		tmp = (((double) M_PI) * l_m) - ((tan(expm1(log1p((((double) M_PI) * l_m)))) / F) / F);
	} else if (F <= 1.48e-36) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / ((((pow(F, 2.0) / (((double) M_PI) * l_m)) - (t_4 * (l_m * t_0))) + (pow(l_m, 5.0) * (((t_5 - t_2) / (((double) M_PI) / t_0)) + ((t_1 * (pow(((double) M_PI), 3.0) * (0.3333333333333333 * t_3))) + (t_4 * (fma((pow(((double) M_PI), 2.0) * -0.5), t_3, 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))) - (-0.0001984126984126984 * pow(((double) M_PI), 7.0)))))))) + (pow(l_m, 3.0) * (t_2 - t_5))));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64((F ^ 2.0) / (pi ^ 3.0))
	t_2 = Float64((t_0 ^ 2.0) * t_1)
	t_3 = Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64(t_0 * (pi ^ 2.0)), Float64((pi ^ 5.0) * 0.041666666666666664)))
	t_4 = Float64((F ^ 2.0) / (pi ^ 2.0))
	t_5 = Float64(t_3 * t_4)
	tmp = 0.0
	if (F <= 5.5e-162)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(expm1(log1p(Float64(pi * l_m)))) / F) / F));
	elseif (F <= 1.48e-36)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(Float64(Float64(Float64((F ^ 2.0) / Float64(pi * l_m)) - Float64(t_4 * Float64(l_m * t_0))) + Float64((l_m ^ 5.0) * Float64(Float64(Float64(t_5 - t_2) / Float64(pi / t_0)) + Float64(Float64(t_1 * Float64((pi ^ 3.0) * Float64(0.3333333333333333 * t_3))) + Float64(t_4 * Float64(fma(Float64((pi ^ 2.0) * -0.5), t_3, fma(0.041666666666666664, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 4.0))), Float64((pi ^ 7.0) * -0.001388888888888889))) - Float64(-0.0001984126984126984 * (pi ^ 7.0)))))))) + Float64((l_m ^ 3.0) * Float64(t_2 - t_5)))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(t$95$0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$4), $MachinePrecision]}, N[(l$95$s * If[LessEqual[F, 5.5e-162], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.48e-36], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[(l$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 5.0], $MachinePrecision] * N[(N[(N[(t$95$5 - t$95$2), $MachinePrecision] / N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * t$95$3 + 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] - N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 3.0], $MachinePrecision] * N[(t$95$2 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 5.50000000000000006e-162

   1. Initial program 69.3%

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

      1. associate-*l/ 69.3%

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

      2. *-un-lft-identity 69.3%

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

      3. associate-/r* 76.8%

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

   4. Applied egg-rr 76.8%

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

      1. expm1-log1p-u 64.7%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   6. Applied egg-rr 64.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   if 5.50000000000000006e-162 < F < 1.48e-36

   1. Initial program 69.8%

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

      1. associate-/r/ 69.7%

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

      2. associate-/l* 69.6%

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

      3. clear-num 69.7%

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

      4. add-sqr-sqrt 69.5%

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

      5. sqrt-prod 69.7%

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

      6. sqr-neg 69.7%

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

      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 15.4%

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

      9. associate-/l/ 15.4%

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

      10. clear-num 15.4%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 69.7%

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

      13. sqr-neg 69.7%

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

      14. sqrt-prod 69.6%

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

      15. add-sqr-sqrt 69.7%

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

      16. pow2 69.7%

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

   4. Applied egg-rr 69.7%

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

      1. add-log-exp 16.3%

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

   6. Applied egg-rr 16.3%

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

   7. Taylor expanded in l around 0 93.0%

      \[\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)}} \]

   9. Simplified 93.0%

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

   if 1.48e-36 < F

   1. Initial program 99.3%

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

      1. sqr-neg 99.3%

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

      2. associate-*l/ 99.3%

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

      3. *-lft-identity 99.3%

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

      4. sqr-neg 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.5 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-36}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)\right)\right) + {\ell}^{5} \cdot \left(\frac{\left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{2}} - {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}}}{\frac{\pi}{{\pi}^{3} \cdot 0.3333333333333333}} + \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\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) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right)\right) + {\ell}^{3} \cdot \left({\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}} - \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot {\pi}^{2}, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \frac{{F}^{2}}{{\pi}^{2}}\\ l_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot l_m + \frac{-1}{\left(\frac{{F}^{2}}{\pi \cdot l_m} - t_1 \cdot \left({\pi}^{3} \cdot \left(l_m \cdot 0.3333333333333333\right)\right)\right) + {l_m}^{3} \cdot \left(t_1 \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, t_0, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right) + {t_0}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1 (/ (pow F 2.0) (pow PI 2.0))))
   (*
    l_s
    (if (<= F 5e-162)
      (- (* PI l_m) (/ (/ (tan (expm1 (log1p (* PI l_m)))) F) F))
      (if (<= F 7e-37)
        (+
         (* PI l_m)
         (/
          -1.0
          (+
           (-
            (/ (pow F 2.0) (* PI l_m))
            (* t_1 (* (pow PI 3.0) (* l_m 0.3333333333333333))))
           (*
            (pow l_m 3.0)
            (+
             (*
              t_1
              (-
               (fma
                (* (pow PI 2.0) -0.5)
                t_0
                (* (pow PI 5.0) 0.041666666666666664))
               (* 0.008333333333333333 (pow PI 5.0))))
             (* (pow t_0 2.0) (/ (pow F 2.0) (pow PI 3.0))))))))
        (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double tmp;
	if (F <= 5e-162) {
		tmp = (((double) M_PI) * l_m) - ((tan(expm1(log1p((((double) M_PI) * l_m)))) / F) / F);
	} else if (F <= 7e-37) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (((pow(F, 2.0) / (((double) M_PI) * l_m)) - (t_1 * (pow(((double) M_PI), 3.0) * (l_m * 0.3333333333333333)))) + (pow(l_m, 3.0) * ((t_1 * (fma((pow(((double) M_PI), 2.0) * -0.5), t_0, (pow(((double) M_PI), 5.0) * 0.041666666666666664)) - (0.008333333333333333 * pow(((double) M_PI), 5.0)))) + (pow(t_0, 2.0) * (pow(F, 2.0) / pow(((double) M_PI), 3.0)))))));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64((F ^ 2.0) / (pi ^ 2.0))
	tmp = 0.0
	if (F <= 5e-162)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(expm1(log1p(Float64(pi * l_m)))) / F) / F));
	elseif (F <= 7e-37)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(Float64(Float64((F ^ 2.0) / Float64(pi * l_m)) - Float64(t_1 * Float64((pi ^ 3.0) * Float64(l_m * 0.3333333333333333)))) + Float64((l_m ^ 3.0) * Float64(Float64(t_1 * Float64(fma(Float64((pi ^ 2.0) * -0.5), t_0, Float64((pi ^ 5.0) * 0.041666666666666664)) - Float64(0.008333333333333333 * (pi ^ 5.0)))) + Float64((t_0 ^ 2.0) * Float64((F ^ 2.0) / (pi ^ 3.0))))))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[F, 5e-162], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e-37], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 3.0], $MachinePrecision] * N[(N[(t$95$1 * N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0 + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 5.00000000000000014e-162

   1. Initial program 69.3%

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

      1. associate-*l/ 69.3%

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

      2. *-un-lft-identity 69.3%

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

      3. associate-/r* 76.8%

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

   4. Applied egg-rr 76.8%

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

      1. expm1-log1p-u 64.7%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   6. Applied egg-rr 64.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]

   if 5.00000000000000014e-162 < F < 7.0000000000000003e-37

   1. Initial program 69.8%

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

      1. associate-/r/ 69.7%

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

      2. associate-/l* 69.6%

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

      3. clear-num 69.7%

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

      4. add-sqr-sqrt 69.5%

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

      5. sqrt-prod 69.7%

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

      6. sqr-neg 69.7%

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

      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 15.4%

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

      9. associate-/l/ 15.4%

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

      10. clear-num 15.4%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 69.7%

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

      13. sqr-neg 69.7%

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

      14. sqrt-prod 69.6%

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

      15. add-sqr-sqrt 69.7%

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

      16. pow2 69.7%

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in l around 0 87.6%

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

   7. Simplified 87.6%

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

   if 7.0000000000000003e-37 < F

   1. Initial program 99.3%

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

      1. sqr-neg 99.3%

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

      2. associate-*l/ 99.3%

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

      3. *-lft-identity 99.3%

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

      4. sqr-neg 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\frac{{F}^{2}}{\pi \cdot \ell} - \frac{{F}^{2}}{{\pi}^{2}} \cdot \left({\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right) + {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} \cdot \frac{{F}^{2}}{{\pi}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 6 \cdot 10^{-135}:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{l_m}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 6e-135)
    (- (* PI l_m) (/ (/ l_m (/ F PI)) F))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 6e-135) {
		tmp = (((double) M_PI) * l_m) - ((l_m / (F / ((double) M_PI))) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 6e-135) {
		tmp = (Math.PI * l_m) - ((l_m / (F / Math.PI)) / F);
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 6e-135:
		tmp = (math.pi * l_m) - ((l_m / (F / math.pi)) / F)
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 6e-135)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / Float64(F / pi)) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 6e-135)
		tmp = (pi * l_m) - ((l_m / (F / pi)) / F);
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 6e-135], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 6.00000000000000024e-135

   1. Initial program 81.1%

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

      1. associate-*l/ 81.2%

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

      2. *-un-lft-identity 81.2%

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

      3. associate-/r* 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in l around 0 81.7%

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

      1. associate-/l* 81.7%

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

   7. Simplified 81.7%

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

   if 6.00000000000000024e-135 < (*.f64 (PI.f64) l)

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. associate-*l/ 74.4%

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

      3. *-lft-identity 74.4%

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

      4. sqr-neg 74.4%

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

   3. Simplified 74.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 6 \cdot 10^{-135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

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) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

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) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

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) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
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) - ((tan((pi * l_m)) / F) / F));
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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. associate-/r* 83.0%

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

4. Applied egg-rr 83.0%

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

5. Final simplification 83.0%

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

Alternative 8: 76.0% accurate, 6.6× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 0.5)
    (- (* PI l_m) (/ (* l_m (/ PI F)) F))
    (- (* PI l_m) (/ (/ (* PI l_m) (- F)) F)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 83.9%

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

      1. sqr-neg 83.9%

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

      2. associate-*l/ 83.9%

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

      3. *-lft-identity 83.9%

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

      4. sqr-neg 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in l around 0 78.7%

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

      1. *-commutative 78.7%

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

      2. times-frac 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l/ 84.4%

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

   9. Applied egg-rr 84.4%

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

   if 0.5 < l

   1. Initial program 62.7%

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

      1. associate-*l/ 62.7%

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

      2. *-un-lft-identity 62.7%

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

      3. associate-/r* 62.7%

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

   4. Applied egg-rr 62.7%

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

      1. frac-2neg 62.7%

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

      2. add-sqr-sqrt 33.5%

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

      3. sqrt-unprod 61.5%

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

      4. sqr-neg 61.5%

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

      5. sqrt-prod 27.9%

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

      6. add-sqr-sqrt 61.2%

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

      7. distribute-neg-frac 61.2%

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

      8. neg-sub0 61.2%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{0 - \frac{\tan \left(\pi \cdot \ell\right)}{F}}}{F} \]

   6. Applied egg-rr 61.2%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{0 - \frac{\tan \left(\pi \cdot \ell\right)}{F}}}{F} \]
   7. Step-by-step derivation

      1. neg-sub0 61.2%

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

      2. neg-mul-1 61.2%

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

      3. metadata-eval 61.2%

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

      4. times-frac 61.2%

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

      5. *-lft-identity 61.2%

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

      6. *-commutative 61.2%

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

      7. mul-1-neg 61.2%

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

   8. Simplified 61.2%

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

   9. Taylor expanded in l around 0 51.1%

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

4. Final simplification 76.3%

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

Alternative 9: 74.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\pi}{F} \cdot \frac{l_m}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ PI F) (/ l_m F)))))

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) - ((((double) M_PI) / F) * (l_m / F)));
}

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) - ((Math.PI / F) * (l_m / F)));
}

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) - ((math.pi / F) * (l_m / F)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F))))
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) - ((pi / F) * (l_m / F)));
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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. sqr-neg 78.7%

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

   2. associate-*l/ 78.8%

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

   3. *-lft-identity 78.8%

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

   4. sqr-neg 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in l around 0 71.0%

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

   1. *-commutative 71.0%

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

   2. times-frac 75.3%

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

7. Applied egg-rr 75.3%

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

8. Final simplification 75.3%

   \[\leadsto \pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F} \]
9. Add Preprocessing

Alternative 10: 74.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{l_m \cdot \frac{\pi}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (* l_m (/ PI F)) F))))

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) - ((l_m * (((double) M_PI) / F)) / F));
}

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) - ((l_m * (Math.PI / F)) / F));
}

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) - ((l_m * (math.pi / F)) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / F)))
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) - ((l_m * (pi / F)) / F));
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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. sqr-neg 78.7%

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

   2. associate-*l/ 78.8%

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

   3. *-lft-identity 78.8%

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

   4. sqr-neg 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in l around 0 71.0%

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

   1. *-commutative 71.0%

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

   2. times-frac 75.3%

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

7. Applied egg-rr 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-*l/ 75.3%

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

9. Applied egg-rr 75.3%

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

10. Final simplification 75.3%

    \[\leadsto \pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024022 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))