VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 87.2%

Time: 34.0s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. F = 9.752513847031436e-291
  2. l = -2037037348561148.5

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \frac{t_0}{\pi}\\ t_2 := \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot t_1\right)\\ t_3 := {\pi}^{3} \cdot -0.16666666666666666\\ t_4 := \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, t_3\right)\\ t_5 := 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{5}, 0.041666666666666664, t_3 \cdot {\pi}^{2}\right)\\ \mathbf{if}\;F \leq 1.45 \cdot 10^{-158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, t_4 - t_2, \mathsf{fma}\left({\ell}^{5}, \left(\mathsf{fma}\left(-0.5, t_5, \mathsf{fma}\left(0.041666666666666664 \cdot {\pi}^{2}, t_0, {\pi}^{5} \cdot -0.001388888888888889\right)\right) - \mathsf{fma}\left({\pi}^{5}, -0.0001984126984126984, t_5 \cdot \frac{\pi \cdot -0.3333333333333333}{\pi}\right)\right) + t_1 \cdot \left(t_2 - t_4\right), \frac{1}{\pi \cdot \ell}\right)\right)\right) \cdot \frac{F}{\frac{1}{F}}}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1 (/ t_0 PI))
        (t_2
         (fma
          (pow PI 3.0)
          0.008333333333333333
          (* (* PI -0.3333333333333333) t_1)))
        (t_3 (* (pow PI 3.0) -0.16666666666666666))
        (t_4 (fma (pow PI 3.0) 0.041666666666666664 t_3))
        (t_5
         (-
          (* 0.008333333333333333 (pow PI 5.0))
          (fma (pow PI 5.0) 0.041666666666666664 (* t_3 (pow PI 2.0))))))
   (if (<= F 1.45e-158)
     (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
     (+
      (* PI l)
      (/
       -1.0
       (*
        (fma
         l
         (* PI -0.3333333333333333)
         (fma
          (pow l 3.0)
          (- t_4 t_2)
          (fma
           (pow l 5.0)
           (+
            (-
             (fma
              -0.5
              t_5
              (fma
               (* 0.041666666666666664 (pow PI 2.0))
               t_0
               (* (pow PI 5.0) -0.001388888888888889)))
             (fma
              (pow PI 5.0)
              -0.0001984126984126984
              (* t_5 (/ (* PI -0.3333333333333333) PI))))
            (* t_1 (- t_2 t_4)))
           (/ 1.0 (* PI l)))))
        (/ F (/ 1.0 F))))))))

C

F = abs(F);
double code(double F, double l) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = t_0 / ((double) M_PI);
	double t_2 = fma(pow(((double) M_PI), 3.0), 0.008333333333333333, ((((double) M_PI) * -0.3333333333333333) * t_1));
	double t_3 = pow(((double) M_PI), 3.0) * -0.16666666666666666;
	double t_4 = fma(pow(((double) M_PI), 3.0), 0.041666666666666664, t_3);
	double t_5 = (0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(pow(((double) M_PI), 5.0), 0.041666666666666664, (t_3 * pow(((double) M_PI), 2.0)));
	double tmp;
	if (F <= 1.45e-158) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) + (-1.0 / (fma(l, (((double) M_PI) * -0.3333333333333333), fma(pow(l, 3.0), (t_4 - t_2), fma(pow(l, 5.0), ((fma(-0.5, t_5, fma((0.041666666666666664 * pow(((double) M_PI), 2.0)), t_0, (pow(((double) M_PI), 5.0) * -0.001388888888888889))) - fma(pow(((double) M_PI), 5.0), -0.0001984126984126984, (t_5 * ((((double) M_PI) * -0.3333333333333333) / ((double) M_PI))))) + (t_1 * (t_2 - t_4))), (1.0 / (((double) M_PI) * l))))) * (F / (1.0 / F))));
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64(t_0 / pi)
	t_2 = fma((pi ^ 3.0), 0.008333333333333333, Float64(Float64(pi * -0.3333333333333333) * t_1))
	t_3 = Float64((pi ^ 3.0) * -0.16666666666666666)
	t_4 = fma((pi ^ 3.0), 0.041666666666666664, t_3)
	t_5 = Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma((pi ^ 5.0), 0.041666666666666664, Float64(t_3 * (pi ^ 2.0))))
	tmp = 0.0
	if (F <= 1.45e-158)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	else
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(fma(l, Float64(pi * -0.3333333333333333), fma((l ^ 3.0), Float64(t_4 - t_2), fma((l ^ 5.0), Float64(Float64(fma(-0.5, t_5, fma(Float64(0.041666666666666664 * (pi ^ 2.0)), t_0, Float64((pi ^ 5.0) * -0.001388888888888889))) - fma((pi ^ 5.0), -0.0001984126984126984, Float64(t_5 * Float64(Float64(pi * -0.3333333333333333) / pi)))) + Float64(t_1 * Float64(t_2 - t_4))), Float64(1.0 / Float64(pi * l))))) * Float64(F / Float64(1.0 / F)))));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.008333333333333333 + N[(N[(Pi * -0.3333333333333333), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.041666666666666664 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664 + N[(t$95$3 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 1.45e-158], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(l * N[(Pi * -0.3333333333333333), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(t$95$4 - t$95$2), $MachinePrecision] + N[(N[Power[l, 5.0], $MachinePrecision] * N[(N[(N[(-0.5 * t$95$5 + N[(N[(0.041666666666666664 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Power[Pi, 5.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 5.0], $MachinePrecision] * -0.0001984126984126984 + N[(t$95$5 * N[(N[(Pi * -0.3333333333333333), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$2 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F / N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if F < 1.4499999999999999e-158

   1. Initial program 66.9%

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

      1. associate-*l/ 67.7%

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

      2. *-un-lft-identity 67.7%

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

      3. associate-/r* 75.7%

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

   3. Applied egg-rr 75.7%

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

      1. add-sqr-sqrt 75.7%

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

      2. pow2 75.7%

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

   5. Applied egg-rr 75.7%

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

   if 1.4499999999999999e-158 < F

   1. Initial program 85.7%

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

      1. associate-/r/ 85.8%

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

      2. associate-/l* 85.8%

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

      3. clear-num 85.8%

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

      4. add-sqr-sqrt 85.7%

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

      5. sqrt-prod 85.8%

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

      6. sqr-neg 85.8%

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

      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 68.2%

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

      9. associate-/r* 68.2%

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

      10. clear-num 68.2%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 85.8%

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

      13. sqr-neg 85.8%

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

      14. sqrt-prod 85.7%

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

      15. add-sqr-sqrt 85.8%

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

      16. pow2 85.8%

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

   3. Applied egg-rr 85.8%

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

      1. pow2 85.8%

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

      2. associate-/l* 85.8%

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

      3. *-un-lft-identity 85.8%

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

      4. div-inv 85.7%

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

      5. times-frac 85.7%

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

   5. Applied egg-rr 85.7%

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

   6. Taylor expanded in l around 0 98.4%

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

   8. Simplified 98.4%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, -0.16666666666666666 \cdot {\pi}^{3}\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right), \mathsf{fma}\left({\ell}^{5}, \left(\mathsf{fma}\left(-0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{5}, 0.041666666666666664, \left(-0.16666666666666666 \cdot {\pi}^{3}\right) \cdot {\pi}^{2}\right), \mathsf{fma}\left(0.041666666666666664 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot -0.001388888888888889\right)\right) - \mathsf{fma}\left({\pi}^{5}, -0.0001984126984126984, \frac{\pi \cdot -0.3333333333333333}{\pi} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{5}, 0.041666666666666664, \left(-0.16666666666666666 \cdot {\pi}^{3}\right) \cdot {\pi}^{2}\right)\right)\right)\right) - \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi} \cdot \left(\mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, -0.16666666666666666 \cdot {\pi}^{3}\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right)\right), \frac{1}{\ell \cdot \pi}\right)\right)\right)} \cdot \frac{F}{\frac{1}{F}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.45 \cdot 10^{-158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, {\pi}^{3} \cdot -0.16666666666666666\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right), \mathsf{fma}\left({\ell}^{5}, \left(\mathsf{fma}\left(-0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{5}, 0.041666666666666664, \left({\pi}^{3} \cdot -0.16666666666666666\right) \cdot {\pi}^{2}\right), \mathsf{fma}\left(0.041666666666666664 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot -0.001388888888888889\right)\right) - \mathsf{fma}\left({\pi}^{5}, -0.0001984126984126984, \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{5}, 0.041666666666666664, \left({\pi}^{3} \cdot -0.16666666666666666\right) \cdot {\pi}^{2}\right)\right) \cdot \frac{\pi \cdot -0.3333333333333333}{\pi}\right)\right) + \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi} \cdot \left(\mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right) - \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, {\pi}^{3} \cdot -0.16666666666666666\right)\right), \frac{1}{\pi \cdot \ell}\right)\right)\right) \cdot \frac{F}{\frac{1}{F}}}\\ \end{array} \]

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, {\pi}^{3} \cdot -0.16666666666666666\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right), \frac{1}{\pi \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 1.65e-158)
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
   (+
    (* PI l)
    (/
     -1.0
     (*
      (/ F (/ 1.0 F))
      (fma
       l
       (* PI -0.3333333333333333)
       (fma
        (pow l 3.0)
        (-
         (fma
          (pow PI 3.0)
          0.041666666666666664
          (* (pow PI 3.0) -0.16666666666666666))
         (fma
          (pow PI 3.0)
          0.008333333333333333
          (*
           (* PI -0.3333333333333333)
           (/ (* (pow PI 3.0) 0.3333333333333333) PI))))
        (/ 1.0 (* PI l)))))))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (F <= 1.65e-158) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) + (-1.0 / ((F / (1.0 / F)) * fma(l, (((double) M_PI) * -0.3333333333333333), fma(pow(l, 3.0), (fma(pow(((double) M_PI), 3.0), 0.041666666666666664, (pow(((double) M_PI), 3.0) * -0.16666666666666666)) - fma(pow(((double) M_PI), 3.0), 0.008333333333333333, ((((double) M_PI) * -0.3333333333333333) * ((pow(((double) M_PI), 3.0) * 0.3333333333333333) / ((double) M_PI))))), (1.0 / (((double) M_PI) * l))))));
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (F <= 1.65e-158)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	else
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64(F / Float64(1.0 / F)) * fma(l, Float64(pi * -0.3333333333333333), fma((l ^ 3.0), Float64(fma((pi ^ 3.0), 0.041666666666666664, Float64((pi ^ 3.0) * -0.16666666666666666)) - fma((pi ^ 3.0), 0.008333333333333333, Float64(Float64(pi * -0.3333333333333333) * Float64(Float64((pi ^ 3.0) * 0.3333333333333333) / pi)))), Float64(1.0 / Float64(pi * l)))))));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[F, 1.65e-158], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(F / N[(1.0 / F), $MachinePrecision]), $MachinePrecision] * N[(l * N[(Pi * -0.3333333333333333), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.041666666666666664 + N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.008333333333333333 + N[(N[(Pi * -0.3333333333333333), $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.6500000000000001e-158

   1. Initial program 66.9%

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

      1. associate-*l/ 67.7%

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

      2. *-un-lft-identity 67.7%

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

      3. associate-/r* 75.7%

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

   3. Applied egg-rr 75.7%

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

      1. add-sqr-sqrt 75.7%

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

      2. pow2 75.7%

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

   5. Applied egg-rr 75.7%

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

   if 1.6500000000000001e-158 < F

   1. Initial program 85.7%

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

      1. associate-/r/ 85.8%

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

      2. associate-/l* 85.8%

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

      3. clear-num 85.8%

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

      4. add-sqr-sqrt 85.7%

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

      5. sqrt-prod 85.8%

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

      6. sqr-neg 85.8%

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

      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 68.2%

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

      9. associate-/r* 68.2%

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

      10. clear-num 68.2%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 85.8%

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

      13. sqr-neg 85.8%

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

      14. sqrt-prod 85.7%

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

      15. add-sqr-sqrt 85.8%

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

      16. pow2 85.8%

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

   3. Applied egg-rr 85.8%

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

      1. pow2 85.8%

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

      2. associate-/l* 85.8%

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

      3. *-un-lft-identity 85.8%

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

      4. div-inv 85.7%

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

      5. times-frac 85.7%

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

   5. Applied egg-rr 85.7%

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

   6. Taylor expanded in l around 0 98.4%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(\ell \cdot \left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) + \left({\ell}^{3} \cdot \left(\left(-0.5 \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right) + 0.041666666666666664 \cdot {\pi}^{3}\right) - \left(0.008333333333333333 \cdot {\pi}^{3} + \frac{\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{\pi}\right)\right) + \frac{1}{\ell \cdot \pi}\right)\right)} \cdot \frac{F}{\frac{1}{F}}} \]
   7. Step-by-step derivation

      1. fma-def 98.4%

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

      2. distribute-rgt-out-- 98.4%

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

      3. metadata-eval 98.4%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \pi \cdot \color{blue}{-0.3333333333333333}, {\ell}^{3} \cdot \left(\left(-0.5 \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right) + 0.041666666666666664 \cdot {\pi}^{3}\right) - \left(0.008333333333333333 \cdot {\pi}^{3} + \frac{\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{\pi}\right)\right) + \frac{1}{\ell \cdot \pi}\right) \cdot \frac{F}{\frac{1}{F}}} \]

      4. fma-def 98.4%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \color{blue}{\mathsf{fma}\left({\ell}^{3}, \left(-0.5 \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right) + 0.041666666666666664 \cdot {\pi}^{3}\right) - \left(0.008333333333333333 \cdot {\pi}^{3} + \frac{\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{\pi}\right), \frac{1}{\ell \cdot \pi}\right)}\right) \cdot \frac{F}{\frac{1}{F}}} \]

   8. Simplified 98.4%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, -0.16666666666666666 \cdot {\pi}^{3}\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right), \frac{1}{\ell \cdot \pi}\right)\right)} \cdot \frac{F}{\frac{1}{F}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \mathsf{fma}\left({\ell}^{3}, \mathsf{fma}\left({\pi}^{3}, 0.041666666666666664, {\pi}^{3} \cdot -0.16666666666666666\right) - \mathsf{fma}\left({\pi}^{3}, 0.008333333333333333, \left(\pi \cdot -0.3333333333333333\right) \cdot \frac{{\pi}^{3} \cdot 0.3333333333333333}{\pi}\right), \frac{1}{\pi \cdot \ell}\right)\right)}\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\pi \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= (/ 1.0 (* F F)) 2e+298)
   (+
    (* PI l)
    (/
     -1.0
     (* (/ F (/ 1.0 F)) (fma l (* PI -0.3333333333333333) (/ 1.0 (* PI l))))))
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((1.0 / (F * F)) <= 2e+298) {
		tmp = (((double) M_PI) * l) + (-1.0 / ((F / (1.0 / F)) * fma(l, (((double) M_PI) * -0.3333333333333333), (1.0 / (((double) M_PI) * l)))));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (Float64(1.0 / Float64(F * F)) <= 2e+298)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64(F / Float64(1.0 / F)) * fma(l, Float64(pi * -0.3333333333333333), Float64(1.0 / Float64(pi * l))))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision], 2e+298], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(F / N[(1.0 / F), $MachinePrecision]), $MachinePrecision] * N[(l * N[(Pi * -0.3333333333333333), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 (*.f64 F F)) < 1.9999999999999999e298

   1. Initial program 89.7%

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

      1. associate-/r/ 89.7%

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

      2. associate-/l* 89.7%

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

      3. clear-num 89.7%

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

      4. add-sqr-sqrt 39.4%

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

      5. sqrt-prod 82.6%

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

      6. sqr-neg 82.6%

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

      7. sqrt-unprod 43.2%

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

      8. add-sqr-sqrt 74.6%

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

      9. associate-/r* 74.6%

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

      10. clear-num 74.6%

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

      11. add-sqr-sqrt 43.2%

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

      12. sqrt-unprod 82.6%

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

      13. sqr-neg 82.6%

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

      14. sqrt-prod 39.4%

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

      15. add-sqr-sqrt 89.7%

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

      16. pow2 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. pow2 89.7%

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

      2. associate-/l* 89.7%

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

      3. *-un-lft-identity 89.7%

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

      4. div-inv 89.7%

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

      5. times-frac 89.7%

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

   5. Applied egg-rr 89.7%

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

   6. Taylor expanded in l around 0 96.5%

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

      1. fma-def 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, -0.5 \cdot \pi - -0.16666666666666666 \cdot \pi, \frac{1}{\ell \cdot \pi}\right)} \cdot \frac{F}{\frac{1}{F}}} \]

      2. distribute-rgt-out-- 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \color{blue}{\pi \cdot \left(-0.5 - -0.16666666666666666\right)}, \frac{1}{\ell \cdot \pi}\right) \cdot \frac{F}{\frac{1}{F}}} \]

      3. metadata-eval 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \pi \cdot \color{blue}{-0.3333333333333333}, \frac{1}{\ell \cdot \pi}\right) \cdot \frac{F}{\frac{1}{F}}} \]

   8. Simplified 96.5%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\ell \cdot \pi}\right)} \cdot \frac{F}{\frac{1}{F}}} \]

   if 1.9999999999999999e298 < (/.f64 1 (*.f64 F F))

   1. Initial program 28.6%

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

      1. associate-*l/ 30.6%

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

      2. *-un-lft-identity 30.6%

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

      3. associate-/r* 50.3%

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

   3. Applied egg-rr 50.3%

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

      1. add-sqr-sqrt 50.5%

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

      2. pow2 50.5%

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

   5. Applied egg-rr 50.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\pi \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \end{array} \]

Alternative 4: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\pi \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= (/ 1.0 (* F F)) 2e+298)
   (+
    (* PI l)
    (/
     -1.0
     (* (/ F (/ 1.0 F)) (fma l (* PI -0.3333333333333333) (/ 1.0 (* PI l))))))
   (- (* PI l) (/ 1.0 (/ F (/ (tan (* PI l)) F))))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((1.0 / (F * F)) <= 2e+298) {
		tmp = (((double) M_PI) * l) + (-1.0 / ((F / (1.0 / F)) * fma(l, (((double) M_PI) * -0.3333333333333333), (1.0 / (((double) M_PI) * l)))));
	} else {
		tmp = (((double) M_PI) * l) - (1.0 / (F / (tan((((double) M_PI) * l)) / F)));
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (Float64(1.0 / Float64(F * F)) <= 2e+298)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64(F / Float64(1.0 / F)) * fma(l, Float64(pi * -0.3333333333333333), Float64(1.0 / Float64(pi * l))))));
	else
		tmp = Float64(Float64(pi * l) - Float64(1.0 / Float64(F / Float64(tan(Float64(pi * l)) / F))));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision], 2e+298], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(F / N[(1.0 / F), $MachinePrecision]), $MachinePrecision] * N[(l * N[(Pi * -0.3333333333333333), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(F / N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 (*.f64 F F)) < 1.9999999999999999e298

   1. Initial program 89.7%

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

      1. associate-/r/ 89.7%

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

      2. associate-/l* 89.7%

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

      3. clear-num 89.7%

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

      4. add-sqr-sqrt 39.4%

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

      5. sqrt-prod 82.6%

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

      6. sqr-neg 82.6%

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

      7. sqrt-unprod 43.2%

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

      8. add-sqr-sqrt 74.6%

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

      9. associate-/r* 74.6%

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

      10. clear-num 74.6%

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

      11. add-sqr-sqrt 43.2%

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

      12. sqrt-unprod 82.6%

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

      13. sqr-neg 82.6%

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

      14. sqrt-prod 39.4%

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

      15. add-sqr-sqrt 89.7%

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

      16. pow2 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. pow2 89.7%

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

      2. associate-/l* 89.7%

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

      3. *-un-lft-identity 89.7%

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

      4. div-inv 89.7%

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

      5. times-frac 89.7%

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

   5. Applied egg-rr 89.7%

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

   6. Taylor expanded in l around 0 96.5%

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

      1. fma-def 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, -0.5 \cdot \pi - -0.16666666666666666 \cdot \pi, \frac{1}{\ell \cdot \pi}\right)} \cdot \frac{F}{\frac{1}{F}}} \]

      2. distribute-rgt-out-- 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \color{blue}{\pi \cdot \left(-0.5 - -0.16666666666666666\right)}, \frac{1}{\ell \cdot \pi}\right) \cdot \frac{F}{\frac{1}{F}}} \]

      3. metadata-eval 96.5%

         \[\leadsto \pi \cdot \ell - \frac{1}{\mathsf{fma}\left(\ell, \pi \cdot \color{blue}{-0.3333333333333333}, \frac{1}{\ell \cdot \pi}\right) \cdot \frac{F}{\frac{1}{F}}} \]

   8. Simplified 96.5%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\ell \cdot \pi}\right)} \cdot \frac{F}{\frac{1}{F}}} \]

   if 1.9999999999999999e298 < (/.f64 1 (*.f64 F F))

   1. Initial program 28.6%

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

      1. associate-/r/ 30.6%

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

      2. associate-/l* 50.3%

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

      3. clear-num 50.3%

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

      4. add-sqr-sqrt 23.7%

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

      5. sqrt-prod 18.2%

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

      6. sqr-neg 18.2%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.7%

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

      9. associate-/r* 1.7%

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

      10. clear-num 1.7%

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

      11. add-sqr-sqrt 0.8%

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

      12. sqrt-unprod 18.2%

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

      13. sqr-neg 18.2%

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

      14. sqrt-prod 17.4%

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

      15. add-sqr-sqrt 30.6%

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

      16. pow2 30.6%

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

   3. Applied egg-rr 30.6%

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

      1. pow2 30.6%

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

      2. associate-/l* 50.3%

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

      3. *-un-lft-identity 50.3%

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

      4. div-inv 50.2%

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

      5. times-frac 30.6%

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

   5. Applied egg-rr 30.6%

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

      1. frac-times 50.2%

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

      2. *-un-lft-identity 50.2%

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

      3. un-div-inv 50.3%

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

   7. Applied egg-rr 50.3%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{1}{F}} \cdot \mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{1}{\pi \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-316}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{\frac{F}{\frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 4e-316)
   (- (* PI l) (/ PI (/ F (/ l F))))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if ((F * F) <= 4e-316) {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F / (l / F)));
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	}
	return tmp;
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	double tmp;
	if ((F * F) <= 4e-316) {
		tmp = (Math.PI * l) - (Math.PI / (F / (l / F)));
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

F = abs(F)
def code(F, l):
	tmp = 0
	if (F * F) <= 4e-316:
		tmp = (math.pi * l) - (math.pi / (F / (l / F)))
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 4e-316)
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F / Float64(l / F))));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

MATLAB

F = abs(F)
function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((F * F) <= 4e-316)
		tmp = (pi * l) - (pi / (F / (l / F)));
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 4e-316], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F / N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 3.999999984e-316

   1. Initial program 27.6%

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

      1. sqr-neg 27.6%

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

      2. associate-*l/ 29.6%

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

      3. *-lft-identity 29.6%

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

      4. sqr-neg 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in l around 0 28.5%

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

      1. *-commutative 28.5%

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

      2. times-frac 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. associate-*l/ 48.4%

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

      2. associate-/l* 48.6%

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

   8. Applied egg-rr 48.6%

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

   if 3.999999984e-316 < (*.f64 F F)

   1. Initial program 89.8%

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

      1. sqr-neg 89.8%

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

      2. associate-*l/ 89.8%

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

      3. *-lft-identity 89.8%

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

      4. sqr-neg 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 4 \cdot 10^{-316}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{\frac{F}{\frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 6: 82.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (tan (* PI l)) F) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. associate-*l/ 73.8%

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

   2. *-un-lft-identity 73.8%

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

   3. associate-/r* 79.1%

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

3. Applied egg-rr 79.1%

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

4. Final simplification 79.1%

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

Alternative 7: 74.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (fma PI l (/ (/ (* PI l) (- F)) F)))

C

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

Julia

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(Pi * l + N[(N[(N[(Pi * l), $MachinePrecision] / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. *-commutative 73.2%

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

   2. sqr-neg 73.2%

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

   3. *-commutative 73.2%

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

   4. fma-neg 73.2%

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

   5. associate-*l/ 73.8%

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

   6. times-frac 79.1%

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

   7. distribute-lft-neg-in 79.1%

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

   8. neg-mul-1 79.1%

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

   9. associate-/r* 79.1%

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

   10. metadata-eval 79.1%

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

   11. distribute-neg-frac 79.1%

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

   12. metadata-eval 79.1%

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

   13. times-frac 73.8%

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

3. Simplified 79.1%

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

4. Taylor expanded in l around 0 73.4%

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

5. Final simplification 73.4%

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

Alternative 8: 74.6% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (* (/ l F) (/ PI F))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. sqr-neg 73.2%

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

   2. associate-*l/ 73.8%

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

   3. *-lft-identity 73.8%

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

   4. sqr-neg 73.8%

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

3. Simplified 73.8%

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

4. Taylor expanded in l around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. times-frac 73.4%

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

6. Applied egg-rr 73.4%

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

7. Final simplification 73.4%

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

Alternative 9: 74.6% accurate, 1.5× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (* PI l) F) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. associate-*l/ 73.8%

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

   2. *-un-lft-identity 73.8%

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

   3. associate-/r* 79.1%

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

3. Applied egg-rr 79.1%

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

4. Taylor expanded in l around 0 73.4%

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

5. Final simplification 73.4%

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

Alternative 10: 27.0% accurate, 2.9× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (/ (- PI) (* F (/ F l))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[((-Pi) / N[(F * N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. sqr-neg 73.2%

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

   2. associate-*l/ 73.8%

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

   3. *-lft-identity 73.8%

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

   4. sqr-neg 73.8%

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

3. Simplified 73.8%

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

4. Taylor expanded in l around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. times-frac 73.4%

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

6. Applied egg-rr 73.4%

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

7. Taylor expanded in F around 0 21.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
8. Step-by-step derivation

   1. mul-1-neg 21.0%

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

   2. *-commutative 21.0%

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

   3. associate-/l* 20.9%

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

9. Simplified 20.9%

   \[\leadsto \color{blue}{-\frac{\pi}{\frac{{F}^{2}}{\ell}}} \]
10. Step-by-step derivation

    1. unpow2 20.9%

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

    2. *-un-lft-identity 20.9%

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

    3. times-frac 26.2%

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

    4. /-rgt-identity 26.2%

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

11. Applied egg-rr 26.2%

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

12. Final simplification 26.2%

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

Reproduce

herbie shell --seed 2023313 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))