VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 87.2%

Time: 36.2s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} t_0 := {\pi}^{2} \cdot -0.5\\ t_1 := \frac{{F}^{2}}{{\pi}^{3}}\\ t_2 := 0.008333333333333333 \cdot {\pi}^{5}\\ t_3 := {\pi}^{3} \cdot 0.3333333333333333\\ t_4 := \mathsf{fma}\left(t_0, t_3, {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_5 := t_4 - t_2\\ t_6 := \frac{{F}^{2}}{{\pi}^{2}}\\ t_7 := t_1 \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right) + t_6 \cdot t_5\\ \mathbf{if}\;F \leq 1.62 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot \ell} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{{\pi}^{2}}\right) - {\ell}^{5} \cdot \left(\left(t_6 \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(t_0, t_2 - t_4, \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) + t_1 \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot t_5\right)\right)\right) + t_3 \cdot \frac{t_7}{\pi}\right)\right) + {\ell}^{3} \cdot t_7}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (pow PI 2.0) -0.5))
        (t_1 (/ (pow F 2.0) (pow PI 3.0)))
        (t_2 (* 0.008333333333333333 (pow PI 5.0)))
        (t_3 (* (pow PI 3.0) 0.3333333333333333))
        (t_4 (fma t_0 t_3 (* (pow PI 5.0) 0.041666666666666664)))
        (t_5 (- t_4 t_2))
        (t_6 (/ (pow F 2.0) (pow PI 2.0)))
        (t_7 (+ (* t_1 (* (pow PI 6.0) 0.1111111111111111)) (* t_6 t_5))))
   (if (<= F 1.62e-161)
     (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
     (if (<= F 5e-52)
       (+
        (* PI l)
        (/
         -1.0
         (+
          (-
           (-
            (/ (pow F 2.0) (* PI l))
            (*
             (pow F 2.0)
             (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (pow PI 2.0))))
           (*
            (pow l 5.0)
            (+
             (+
              (*
               t_6
               (-
                (* -0.0001984126984126984 (pow PI 7.0))
                (fma
                 t_0
                 (- t_2 t_4)
                 (fma
                  0.041666666666666664
                  (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 4.0)))
                  (* (pow PI 7.0) -0.001388888888888889)))))
              (* t_1 (* (pow PI 3.0) (* 0.3333333333333333 t_5))))
             (* t_3 (/ t_7 PI)))))
          (* (pow l 3.0) t_7))))
       (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) F))))))

C

F = abs(F);
double code(double F, double l) {
	double t_0 = pow(((double) M_PI), 2.0) * -0.5;
	double t_1 = pow(F, 2.0) / pow(((double) M_PI), 3.0);
	double t_2 = 0.008333333333333333 * pow(((double) M_PI), 5.0);
	double t_3 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_4 = fma(t_0, t_3, (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_5 = t_4 - t_2;
	double t_6 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double t_7 = (t_1 * (pow(((double) M_PI), 6.0) * 0.1111111111111111)) + (t_6 * t_5);
	double tmp;
	if (F <= 1.62e-161) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	} else if (F <= 5e-52) {
		tmp = (((double) M_PI) * l) + (-1.0 / ((((pow(F, 2.0) / (((double) M_PI) * l)) - (pow(F, 2.0) * ((pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)) / pow(((double) M_PI), 2.0)))) - (pow(l, 5.0) * (((t_6 * ((-0.0001984126984126984 * pow(((double) M_PI), 7.0)) - fma(t_0, (t_2 - t_4), fma(0.041666666666666664, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 4.0))), (pow(((double) M_PI), 7.0) * -0.001388888888888889))))) + (t_1 * (pow(((double) M_PI), 3.0) * (0.3333333333333333 * t_5)))) + (t_3 * (t_7 / ((double) M_PI)))))) + (pow(l, 3.0) * t_7)));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F) / F);
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	t_0 = Float64((pi ^ 2.0) * -0.5)
	t_1 = Float64((F ^ 2.0) / (pi ^ 3.0))
	t_2 = Float64(0.008333333333333333 * (pi ^ 5.0))
	t_3 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_4 = fma(t_0, t_3, Float64((pi ^ 5.0) * 0.041666666666666664))
	t_5 = Float64(t_4 - t_2)
	t_6 = Float64((F ^ 2.0) / (pi ^ 2.0))
	t_7 = Float64(Float64(t_1 * Float64((pi ^ 6.0) * 0.1111111111111111)) + Float64(t_6 * t_5))
	tmp = 0.0
	if (F <= 1.62e-161)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	elseif (F <= 5e-52)
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(Float64(Float64(Float64((F ^ 2.0) / Float64(pi * l)) - Float64((F ^ 2.0) * Float64(Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)) / (pi ^ 2.0)))) - Float64((l ^ 5.0) * Float64(Float64(Float64(t_6 * Float64(Float64(-0.0001984126984126984 * (pi ^ 7.0)) - fma(t_0, Float64(t_2 - t_4), fma(0.041666666666666664, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 4.0))), Float64((pi ^ 7.0) * -0.001388888888888889))))) + Float64(t_1 * Float64((pi ^ 3.0) * Float64(0.3333333333333333 * t_5)))) + Float64(t_3 * Float64(t_7 / pi))))) + Float64((l ^ 3.0) * t_7))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F) / 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, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$3 + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$1 * N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 1.62e-161], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-52], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[l, 5.0], $MachinePrecision] * N[(N[(N[(t$95$6 * N[(N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(t$95$2 - t$95$4), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(t$95$7 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.62e-161

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

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

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 91.3%

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

   6. Simplified 91.3%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(\left(\frac{{F}^{2}}{\ell \cdot \pi} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)}{{\pi}^{2}}\right) - {\ell}^{5} \cdot \left(\left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right), \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)\right)\right) - \frac{\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right)}{\pi} \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)\right)\right) - {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right)\right)}} \]

   if 5e-52 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.62 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\left(\left(\frac{{F}^{2}}{\pi \cdot \ell} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{{\pi}^{2}}\right) - {\ell}^{5} \cdot \left(\left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{7} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right), \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right)\right) + \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{3} \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)\right)\right) + \left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \frac{\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)}{\pi}\right)\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 86.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(\frac{{F}^{2}}{\pi \cdot \ell} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{{\pi}^{2}}\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-161)
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
   (if (<= F 3e-52)
     (-
      (* PI l)
      (/
       1.0
       (+
        (-
         (/ (pow F 2.0) (* PI l))
         (*
          (pow F 2.0)
          (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (pow PI 2.0))))
        (*
         (pow l 3.0)
         (+
          (* (/ (pow F 2.0) (pow PI 3.0)) (* (pow PI 6.0) 0.1111111111111111))
          (*
           (/ (pow F 2.0) (pow PI 2.0))
           (-
            (fma
             (* (pow PI 2.0) -0.5)
             (* (pow PI 3.0) 0.3333333333333333)
             (* (pow PI 5.0) 0.041666666666666664))
            (* 0.008333333333333333 (pow PI 5.0)))))))))
     (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-161) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	} else if (F <= 3e-52) {
		tmp = (((double) M_PI) * l) - (1.0 / (((pow(F, 2.0) / (((double) M_PI) * l)) - (pow(F, 2.0) * ((pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)) / pow(((double) M_PI), 2.0)))) + (pow(l, 3.0) * (((pow(F, 2.0) / pow(((double) M_PI), 3.0)) * (pow(((double) M_PI), 6.0) * 0.1111111111111111)) + ((pow(F, 2.0) / pow(((double) M_PI), 2.0)) * (fma((pow(((double) M_PI), 2.0) * -0.5), (pow(((double) M_PI), 3.0) * 0.3333333333333333), (pow(((double) M_PI), 5.0) * 0.041666666666666664)) - (0.008333333333333333 * pow(((double) M_PI), 5.0))))))));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F) / F);
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (F <= 1.1e-161)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	elseif (F <= 3e-52)
		tmp = Float64(Float64(pi * l) - Float64(1.0 / Float64(Float64(Float64((F ^ 2.0) / Float64(pi * l)) - Float64((F ^ 2.0) * Float64(Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)) / (pi ^ 2.0)))) + Float64((l ^ 3.0) * Float64(Float64(Float64((F ^ 2.0) / (pi ^ 3.0)) * Float64((pi ^ 6.0) * 0.1111111111111111)) + Float64(Float64((F ^ 2.0) / (pi ^ 2.0)) * Float64(fma(Float64((pi ^ 2.0) * -0.5), Float64((pi ^ 3.0) * 0.3333333333333333), Float64((pi ^ 5.0) * 0.041666666666666664)) - Float64(0.008333333333333333 * (pi ^ 5.0)))))))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F) / F));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[F, 1.1e-161], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e-52], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.10000000000000001e-161

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

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

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 86.8%

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

   6. Simplified 86.8%

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(\frac{{F}^{2}}{\ell \cdot \pi} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5 \cdot {\pi}^{2}, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right)\right)}} \]

   if 3e-52 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-52}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(\frac{{F}^{2}}{\pi \cdot \ell} - {F}^{2} \cdot \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{{\pi}^{2}}\right) + {\ell}^{3} \cdot \left(\frac{{F}^{2}}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right) + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, {\pi}^{3} \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F}\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\mathsf{fma}\left(-1, \frac{\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(\ell \cdot {F}^{2}\right)}{{\pi}^{2}}, \frac{\frac{{F}^{2}}{\ell}}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \tan \left({\left(\sqrt[3]{\pi \cdot \ell}\right)}^{3}\right) \cdot \frac{-1}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 4.5e-171)
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
   (if (<= F 1.85)
     (-
      (* PI l)
      (/
       1.0
       (fma
        -1.0
        (/
         (* (* (pow PI 3.0) 0.3333333333333333) (* l (pow F 2.0)))
         (pow PI 2.0))
        (/ (/ (pow F 2.0) l) PI))))
     (+ (* PI l) (* (tan (pow (cbrt (* PI l)) 3.0)) (/ -1.0 (* F F)))))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (F <= 4.5e-171) {
		tmp = (((double) M_PI) * l) - ((tan((l * pow(sqrt(((double) M_PI)), 2.0))) / F) / F);
	} else if (F <= 1.85) {
		tmp = (((double) M_PI) * l) - (1.0 / fma(-1.0, (((pow(((double) M_PI), 3.0) * 0.3333333333333333) * (l * pow(F, 2.0))) / pow(((double) M_PI), 2.0)), ((pow(F, 2.0) / l) / ((double) M_PI))));
	} else {
		tmp = (((double) M_PI) * l) + (tan(pow(cbrt((((double) M_PI) * l)), 3.0)) * (-1.0 / (F * F)));
	}
	return tmp;
}

Julia

F = abs(F)
function code(F, l)
	tmp = 0.0
	if (F <= 4.5e-171)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * (sqrt(pi) ^ 2.0))) / F) / F));
	elseif (F <= 1.85)
		tmp = Float64(Float64(pi * l) - Float64(1.0 / fma(-1.0, Float64(Float64(Float64((pi ^ 3.0) * 0.3333333333333333) * Float64(l * (F ^ 2.0))) / (pi ^ 2.0)), Float64(Float64((F ^ 2.0) / l) / pi))));
	else
		tmp = Float64(Float64(pi * l) + Float64(tan((cbrt(Float64(pi * l)) ^ 3.0)) * Float64(-1.0 / Float64(F * F))));
	end
	return tmp
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[F, 4.5e-171], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(-1.0 * N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(l * N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[F, 2.0], $MachinePrecision] / l), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[Tan[N[Power[N[Power[N[(Pi * l), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.5000000000000004e-171

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

   if 4.5000000000000004e-171 < F < 1.8500000000000001

   1. Initial program 77.9%

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

      1. associate-/r/ 77.9%

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

      2. associate-/l* 78.0%

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

      3. clear-num 78.0%

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

      4. add-sqr-sqrt 77.8%

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

      5. sqrt-prod 78.0%

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

      6. sqr-neg 78.0%

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

      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 34.4%

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

      9. associate-/r* 34.4%

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

      10. clear-num 34.4%

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

      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 77.9%

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

      13. sqr-neg 77.9%

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

      14. sqrt-prod 77.7%

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

      15. add-sqr-sqrt 77.9%

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

      16. pow2 77.9%

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

   3. Applied egg-rr 77.9%

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

   4. Taylor expanded in l around 0 82.6%

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

      1. fma-def 82.6%

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

      2. associate-*r* 82.6%

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

      3. distribute-rgt-out-- 82.6%

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

      4. metadata-eval 82.6%

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

      5. associate-/r* 82.6%

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

   6. Simplified 82.6%

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

   if 1.8500000000000001 < F

   1. Initial program 99.6%

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

      1. add-cube-cbrt 99.6%

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

      2. pow3 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot {\left(\sqrt{\pi}\right)}^{2}\right)}{F}}{F}\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\mathsf{fma}\left(-1, \frac{\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \left(\ell \cdot {F}^{2}\right)}{{\pi}^{2}}, \frac{\frac{{F}^{2}}{\ell}}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \tan \left({\left(\sqrt[3]{\pi \cdot \ell}\right)}^{3}\right) \cdot \frac{-1}{F \cdot F}\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 4.5e-171)
   (- (* PI l) (/ (/ (tan (* l (pow (sqrt PI) 2.0))) F) F))
   (if (<= F 2e-51)
     (+
      (* PI l)
      (/
       -1.0
       (-
        (/ (pow F 2.0) (* PI l))
        (*
         (pow F 2.0)
         (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (pow PI 2.0))))))
     (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) F)))))

C

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

Java

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

Julia

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := If[LessEqual[F, 4.5e-171], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-51], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(N[(N[Power[F, 2.0], $MachinePrecision] / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.5000000000000004e-171

   1. Initial program 65.5%

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

      1. associate-*l/ 65.6%

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

      2. *-un-lft-identity 65.6%

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

      3. associate-/r* 74.8%

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

   3. Applied egg-rr 74.8%

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

      1. add-sqr-sqrt 74.9%

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

      2. pow2 74.9%

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

   5. Applied egg-rr 74.9%

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

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

   1. Initial program 71.9%

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

      1. associate-/r/ 71.8%

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

      2. associate-/l* 71.9%

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

      3. clear-num 71.9%

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

      4. add-sqr-sqrt 71.8%

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

      5. sqrt-prod 71.9%

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

      6. sqr-neg 71.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 29.7%

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

      9. associate-/r* 29.7%

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

      10. clear-num 29.7%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 71.8%

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

      13. sqr-neg 71.8%

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

      14. sqrt-prod 71.6%

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

      15. add-sqr-sqrt 71.8%

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

      16. pow2 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in l around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. *-lft-identity 77.9%

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

      5. times-frac 77.9%

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

      6. unpow2 77.9%

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

      7. associate-*r/ 77.9%

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

      8. /-rgt-identity 77.9%

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

      9. unpow2 77.9%

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

   6. Simplified 77.9%

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

   if 2e-51 < F

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 81.6%

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

Alternative 5: 82.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

F = abs(F)
function code(F, l)
	return fma(pi, l, Float64(Float64(tan(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[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. sqr-neg 75.0%

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

   3. *-commutative 75.0%

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

   4. fma-neg 75.0%

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

   5. associate-*l/ 75.0%

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

   6. times-frac 81.0%

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

   7. distribute-lft-neg-in 81.0%

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

   8. neg-mul-1 81.0%

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

   9. associate-/r* 81.0%

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

   10. metadata-eval 81.0%

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

   11. distribute-neg-frac 81.0%

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

   12. metadata-eval 81.0%

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

   13. times-frac 75.0%

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

3. Simplified 81.1%

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

4. Final simplification 81.1%

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

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l)
 :precision binary64
 (if (<= F 4.5e-171)
   (- (* PI l) (/ (/ PI F) (/ F l)))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

F = abs(F);
double code(double F, double l) {
	double tmp;
	if (F <= 4.5e-171) {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) / (F / l));
	} 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 <= 4.5e-171) {
		tmp = (Math.PI * l) - ((Math.PI / F) / (F / l));
	} 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 <= 4.5e-171:
		tmp = (math.pi * l) - ((math.pi / F) / (F / l))
	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 (F <= 4.5e-171)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) / Float64(F / l)));
	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 <= 4.5e-171)
		tmp = (pi * l) - ((pi / F) / (F / l));
	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[F, 4.5e-171], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] / N[(F / l), $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 F < 4.5000000000000004e-171

   1. Initial program 65.5%

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

      1. sqr-neg 65.5%

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

      2. associate-*l/ 65.6%

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

      3. *-lft-identity 65.6%

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

      4. sqr-neg 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in l around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. times-frac 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. clear-num 67.6%

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

      2. un-div-inv 67.6%

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

   8. Applied egg-rr 67.6%

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

   if 4.5000000000000004e-171 < F

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. associate-*l/ 93.0%

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

      3. *-lft-identity 93.0%

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

      4. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 7: 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 75.0%

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

   1. associate-*l/ 75.0%

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

   2. *-un-lft-identity 75.0%

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

   3. associate-/r* 81.1%

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

3. Applied egg-rr 81.1%

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

4. Final simplification 81.1%

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

Alternative 8: 75.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-*l/ 75.0%

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

   3. *-lft-identity 75.0%

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

   4. sqr-neg 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. times-frac 72.4%

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

6. Applied egg-rr 72.4%

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

7. Final simplification 72.4%

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

Alternative 9: 75.3% accurate, 1.5× speedup

Math

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

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 / Float64(F / 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 / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-*l/ 75.0%

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

   2. *-un-lft-identity 75.0%

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

   3. associate-/r* 81.1%

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

3. Applied egg-rr 81.1%

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

4. Taylor expanded in l around 0 72.4%

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

   1. associate-/l* 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

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

Alternative 10: 75.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-*l/ 75.0%

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

   2. *-un-lft-identity 75.0%

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

   3. associate-/r* 81.1%

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

3. Applied egg-rr 81.1%

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

4. Taylor expanded in l around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. associate-/l* 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

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

Alternative 11: 21.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \pi \cdot \left(\left(-\ell\right) \cdot {F}^{-2}\right) \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (* PI (* (- l) (pow F -2.0))))

C

F = abs(F);
double code(double F, double l) {
	return ((double) M_PI) * (-l * pow(F, -2.0));
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	return Math.PI * (-l * Math.pow(F, -2.0));
}

Python

F = abs(F)
def code(F, l):
	return math.pi * (-l * math.pow(F, -2.0))

Julia

F = abs(F)
function code(F, l)
	return Float64(pi * Float64(Float64(-l) * (F ^ -2.0)))
end

MATLAB

F = abs(F)
function tmp = code(F, l)
	tmp = pi * (-l * (F ^ -2.0));
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(Pi * N[((-l) * N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. sqr-neg 75.0%

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

   3. *-commutative 75.0%

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

   4. fma-neg 75.0%

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

   5. associate-*l/ 75.0%

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

   6. times-frac 81.0%

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

   7. distribute-lft-neg-in 81.0%

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

   8. neg-mul-1 81.0%

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

   9. associate-/r* 81.0%

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

   10. metadata-eval 81.0%

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

   11. distribute-neg-frac 81.0%

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

   12. metadata-eval 81.0%

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

   13. times-frac 75.0%

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

3. Simplified 81.1%

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

4. Taylor expanded in F around 0 20.8%

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

   1. associate-*r/ 20.8%

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

   2. *-commutative 20.8%

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

   3. neg-mul-1 20.8%

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

   4. *-commutative 20.8%

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

6. Simplified 20.8%

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

7. Taylor expanded in l around 0 19.5%

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

   1. mul-1-neg 19.5%

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

   2. *-commutative 19.5%

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

   3. associate-*l/ 19.1%

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

   4. distribute-rgt-neg-in 19.1%

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

9. Simplified 19.1%

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

    1. distribute-rgt-neg-out 19.1%

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

    2. div-inv 19.1%

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

    3. associate-*l* 19.5%

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

    4. pow-flip 19.5%

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

    5. metadata-eval 19.5%

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

11. Applied egg-rr 19.5%

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

12. Final simplification 19.5%

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

Alternative 12: 21.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} F = |F|\\ \\ \frac{\pi \cdot \left(-\ell\right)}{{F}^{2}} \end{array} \]

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (/ (* PI (- l)) (pow F 2.0)))

C

F = abs(F);
double code(double F, double l) {
	return (((double) M_PI) * -l) / pow(F, 2.0);
}

Java

F = Math.abs(F);
public static double code(double F, double l) {
	return (Math.PI * -l) / Math.pow(F, 2.0);
}

Python

F = abs(F)
def code(F, l):
	return (math.pi * -l) / math.pow(F, 2.0)

Julia

F = abs(F)
function code(F, l)
	return Float64(Float64(pi * Float64(-l)) / (F ^ 2.0))
end

MATLAB

F = abs(F)
function tmp = code(F, l)
	tmp = (pi * -l) / (F ^ 2.0);
end

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(N[(Pi * (-l)), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. sqr-neg 75.0%

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

   3. *-commutative 75.0%

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

   4. fma-neg 75.0%

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

   5. associate-*l/ 75.0%

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

   6. times-frac 81.0%

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

   7. distribute-lft-neg-in 81.0%

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

   8. neg-mul-1 81.0%

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

   9. associate-/r* 81.0%

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

   10. metadata-eval 81.0%

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

   11. distribute-neg-frac 81.0%

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

   12. metadata-eval 81.0%

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

   13. times-frac 75.0%

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

3. Simplified 81.1%

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

4. Taylor expanded in F around 0 20.8%

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

   1. associate-*r/ 20.8%

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

   2. *-commutative 20.8%

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

   3. neg-mul-1 20.8%

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

   4. *-commutative 20.8%

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

6. Simplified 20.8%

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

7. Taylor expanded in l around 0 19.5%

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

   1. mul-1-neg 19.5%

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

   2. *-commutative 19.5%

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

   3. associate-*l/ 19.1%

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

   4. distribute-rgt-neg-in 19.1%

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

9. Simplified 19.1%

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

10. Taylor expanded in F around 0 19.5%

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

    1. associate-*r/ 19.5%

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

    2. *-commutative 19.5%

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

    3. neg-mul-1 19.5%

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

    4. distribute-rgt-neg-in 19.5%

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

12. Simplified 19.5%

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

13. Final simplification 19.5%

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

Alternative 13: 21.2% accurate, 2.9× speedup

Math

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

FPCore

NOTE: F should be positive before calling this function
(FPCore (F l) :precision binary64 (* l (* (/ PI F) (/ -1.0 F))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: F should be positive before calling this function
code[F_, l_] := N[(l * N[(N[(Pi / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. sqr-neg 75.0%

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

   3. *-commutative 75.0%

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

   4. fma-neg 75.0%

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

   5. associate-*l/ 75.0%

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

   6. times-frac 81.0%

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

   7. distribute-lft-neg-in 81.0%

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

   8. neg-mul-1 81.0%

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

   9. associate-/r* 81.0%

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

   10. metadata-eval 81.0%

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

   11. distribute-neg-frac 81.0%

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

   12. metadata-eval 81.0%

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

   13. times-frac 75.0%

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

3. Simplified 81.1%

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

4. Taylor expanded in F around 0 20.8%

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

   1. associate-*r/ 20.8%

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

   2. *-commutative 20.8%

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

   3. neg-mul-1 20.8%

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

   4. *-commutative 20.8%

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

6. Simplified 20.8%

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

7. Taylor expanded in l around 0 19.5%

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

   1. mul-1-neg 19.5%

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

   2. *-commutative 19.5%

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

   3. associate-*l/ 19.1%

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

   4. distribute-rgt-neg-in 19.1%

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

9. Simplified 19.1%

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

    1. *-un-lft-identity 19.1%

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

    2. pow2 19.1%

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

    3. times-frac 19.1%

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

11. Applied egg-rr 19.1%

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

12. Final simplification 19.1%

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

Alternative 14: 4.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. sqr-neg 75.0%

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

   3. *-commutative 75.0%

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

   4. fma-neg 75.0%

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

   5. associate-*l/ 75.0%

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

   6. times-frac 81.0%

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

   7. distribute-lft-neg-in 81.0%

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

   8. neg-mul-1 81.0%

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

   9. associate-/r* 81.0%

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

   10. metadata-eval 81.0%

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

   11. distribute-neg-frac 81.0%

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

   12. metadata-eval 81.0%

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

   13. times-frac 75.0%

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

3. Simplified 81.1%

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

4. Taylor expanded in F around 0 20.8%

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

   1. associate-*r/ 20.8%

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

   2. *-commutative 20.8%

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

   3. neg-mul-1 20.8%

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

   4. *-commutative 20.8%

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

6. Simplified 20.8%

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

7. Taylor expanded in l around 0 19.5%

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

   1. mul-1-neg 19.5%

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

   2. *-commutative 19.5%

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

   3. associate-*l/ 19.1%

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

   4. distribute-rgt-neg-in 19.1%

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

9. Simplified 19.1%

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

    1. associate-*l/ 19.5%

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

    2. pow2 19.5%

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

    3. associate-/r* 25.5%

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

    4. add-sqr-sqrt 14.7%

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

    5. sqrt-unprod 9.8%

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

    6. sqr-neg 9.8%

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

    7. sqrt-unprod 2.3%

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

    8. add-sqr-sqrt 4.3%

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

11. Applied egg-rr 4.3%

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

12. Final simplification 4.3%

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

Reproduce

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