VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.1% → 97.7%

Time: 2.1min

Alternatives: 6

Speedup: 0.7×

• could not determine a ground truth

  1. F = 1.4082137921468174e+166
  2. l = -3077758019909248.5

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: 77.1% 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: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+25) (not (<= (* PI l) 5e-17)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (tan (* PI l)))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+25) || !((((double) M_PI) * l) <= 5e-17)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / tan((((double) M_PI) * l))));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+25) || !((Math.PI * l) <= 5e-17)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / Math.tan((Math.PI * l))));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+25) or not ((math.pi * l) <= 5e-17):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / math.tan((math.pi * l))))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+25) || !(Float64(pi * l) <= 5e-17))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / tan(Float64(pi * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+25) || ~(((pi * l) <= 5e-17)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / tan((pi * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+25], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-17]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2.00000000000000018e25 or 4.9999999999999999e-17 < (*.f64 (PI.f64) l)

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

      2. sqr-neg 59.0%

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

      3. *-commutative 59.0%

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

      4. fma-neg 59.0%

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

      5. associate-*l/ 59.0%

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

      6. times-frac 59.0%

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

      7. distribute-lft-neg-in 59.0%

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

      8. neg-mul-1 59.0%

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

      9. associate-/r* 59.0%

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

      10. metadata-eval 59.0%

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

      11. distribute-neg-frac 59.0%

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

      12. metadata-eval 59.0%

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

      13. times-frac 59.0%

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

   3. Simplified 59.0%

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

      1. fma-udef 59.0%

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

      2. associate-/l/ 59.0%

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

      3. associate-/r* 59.0%

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

      4. add-sqr-sqrt 35.4%

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

      5. sqrt-unprod 59.1%

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

      6. sqr-neg 59.1%

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

      7. sqrt-prod 23.7%

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

      8. add-sqr-sqrt 59.0%

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

      9. associate-/r* 59.0%

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

      10. div-inv 59.0%

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

      11. pow2 59.0%

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

      12. pow-flip 59.0%

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

      13. metadata-eval 59.0%

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

   5. Applied egg-rr 59.0%

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

      1. *-commutative 59.0%

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

      2. fma-def 59.0%

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

      3. *-commutative 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in l around inf 99.7%

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

   if -2.00000000000000018e25 < (*.f64 (PI.f64) l) < 4.9999999999999999e-17

   1. Initial program 87.6%

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

      1. associate-/r/ 87.7%

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

      2. associate-/l* 99.7%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 50.4%

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

      5. sqrt-prod 67.0%

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

      6. sqr-neg 67.0%

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

      7. sqrt-unprod 23.7%

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

      8. add-sqr-sqrt 48.4%

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

      9. div-inv 48.4%

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

      10. metadata-eval 48.4%

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

      11. add-sqr-sqrt 24.7%

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

      12. sqrt-prod 69.0%

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

      13. sqrt-div 69.0%

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

      14. associate-*l/ 69.0%

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

      15. clear-num 69.0%

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

      16. associate-*l/ 69.0%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\pi}^{3} \cdot 0.3333333333333333\\ t_1 := \frac{F}{{\pi}^{3}}\\ t_2 := 0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_3 := \frac{F}{{\pi}^{2}}\\ \pi \cdot \ell + \frac{\frac{-1}{F}}{\left(\left(\frac{F}{\pi \cdot \ell} - \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{\frac{{\pi}^{2}}{F}}\right) + {\ell}^{5} \cdot \left(t_1 \cdot \left(t_2 \cdot t_0\right) + \left(t_0 \cdot \frac{t_3 \cdot t_2 - t_1 \cdot {t_0}^{2}}{\pi} + t_3 \cdot \left(\mathsf{fma}\left({\pi}^{2} \cdot -0.5, t_2, \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right)\right) - \log \left({\left(e^{{\ell}^{3}}\right)}^{\left(t_2 \cdot \left(F \cdot {\pi}^{-2}\right) - \frac{F \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right)}{{\pi}^{3}}\right)}\right)} \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (pow PI 3.0) 0.3333333333333333))
        (t_1 (/ F (pow PI 3.0)))
        (t_2
         (-
          (* 0.008333333333333333 (pow PI 5.0))
          (fma
           -0.5
           (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 2.0)))
           (* (pow PI 5.0) 0.041666666666666664))))
        (t_3 (/ F (pow PI 2.0))))
   (+
    (* PI l)
    (/
     (/ -1.0 F)
     (-
      (+
       (-
        (/ F (* PI l))
        (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (/ (pow PI 2.0) F)))
       (*
        (pow l 5.0)
        (+
         (* t_1 (* t_2 t_0))
         (+
          (* t_0 (/ (- (* t_3 t_2) (* t_1 (pow t_0 2.0))) PI))
          (*
           t_3
           (-
            (fma
             (* (pow PI 2.0) -0.5)
             t_2
             (fma
              0.041666666666666664
              (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 4.0)))
              (* (pow PI 7.0) -0.001388888888888889)))
            (* -0.0001984126984126984 (pow PI 7.0))))))))
      (log
       (pow
        (exp (pow l 3.0))
        (-
         (* t_2 (* F (pow PI -2.0)))
         (/ (* F (* (pow PI 6.0) 0.1111111111111111)) (pow PI 3.0))))))))))

C

double code(double F, double l) {
	double t_0 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_1 = F / pow(((double) M_PI), 3.0);
	double t_2 = (0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 2.0))), (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_3 = F / pow(((double) M_PI), 2.0);
	return (((double) M_PI) * l) + ((-1.0 / F) / ((((F / (((double) M_PI) * l)) - ((pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)) / (pow(((double) M_PI), 2.0) / F))) + (pow(l, 5.0) * ((t_1 * (t_2 * t_0)) + ((t_0 * (((t_3 * t_2) - (t_1 * pow(t_0, 2.0))) / ((double) M_PI))) + (t_3 * (fma((pow(((double) M_PI), 2.0) * -0.5), t_2, fma(0.041666666666666664, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 4.0))), (pow(((double) M_PI), 7.0) * -0.001388888888888889))) - (-0.0001984126984126984 * pow(((double) M_PI), 7.0)))))))) - log(pow(exp(pow(l, 3.0)), ((t_2 * (F * pow(((double) M_PI), -2.0))) - ((F * (pow(((double) M_PI), 6.0) * 0.1111111111111111)) / pow(((double) M_PI), 3.0)))))));
}

Julia

function code(F, l)
	t_0 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_1 = Float64(F / (pi ^ 3.0))
	t_2 = Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 2.0))), Float64((pi ^ 5.0) * 0.041666666666666664)))
	t_3 = Float64(F / (pi ^ 2.0))
	return Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(Float64(Float64(Float64(F / Float64(pi * l)) - Float64(Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)) / Float64((pi ^ 2.0) / F))) + Float64((l ^ 5.0) * Float64(Float64(t_1 * Float64(t_2 * t_0)) + Float64(Float64(t_0 * Float64(Float64(Float64(t_3 * t_2) - Float64(t_1 * (t_0 ^ 2.0))) / pi)) + Float64(t_3 * Float64(fma(Float64((pi ^ 2.0) * -0.5), t_2, fma(0.041666666666666664, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 4.0))), Float64((pi ^ 7.0) * -0.001388888888888889))) - Float64(-0.0001984126984126984 * (pi ^ 7.0)))))))) - log((exp((l ^ 3.0)) ^ Float64(Float64(t_2 * Float64(F * (pi ^ -2.0))) - Float64(Float64(F * Float64((pi ^ 6.0) * 0.1111111111111111)) / (pi ^ 3.0))))))))
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(F / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(F / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 5.0], $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] - N[(t$95$1 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * t$95$2 + N[(0.041666666666666664 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[Power[N[Exp[N[Power[l, 3.0], $MachinePrecision]], $MachinePrecision], N[(N[(t$95$2 * N[(F * N[Power[Pi, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(F * N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 74.6%

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

   1. associate-/r/ 74.7%

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

   2. associate-/l* 81.2%

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

   3. clear-num 81.2%

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

   4. add-sqr-sqrt 38.3%

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

   5. sqrt-prod 63.3%

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

   6. sqr-neg 63.3%

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

   7. sqrt-unprod 29.0%

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

   8. add-sqr-sqrt 53.2%

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

   9. div-inv 53.2%

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

   10. metadata-eval 53.2%

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

   11. add-sqr-sqrt 24.2%

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

   12. sqrt-prod 64.5%

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

   13. sqrt-div 64.5%

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

   14. associate-*l/ 64.5%

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

   15. clear-num 64.5%

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

   16. associate-*l/ 64.5%

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

3. Applied egg-rr 81.2%

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

4. Taylor expanded in l around 0 94.4%

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

6. Simplified 95.6%

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

   1. add-log-exp 94.8%

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

   2. exp-prod 98.1%

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

8. Applied egg-rr 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 96.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\pi}^{5} \cdot 0.13333333333333333\\ \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\left(\left(\frac{F}{\pi \cdot \ell} - \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{\frac{{\pi}^{2}}{F}}\right) + {\ell}^{5} \cdot \left(F \cdot \left(\mathsf{fma}\left(t_0, -0.5, {\pi}^{5} \cdot 0.0125 + 0.3333333333333333 \cdot \mathsf{fma}\left({\pi}^{2}, {\pi}^{3} \cdot 0.008333333333333333 - {\pi}^{3} \cdot -0.013888888888888888, t_0\right)\right) - -0.0001984126984126984 \cdot {\pi}^{5}\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} - \frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (pow PI 5.0) 0.13333333333333333)))
   (if (<= (* PI l) 5e-17)
     (-
      (* PI l)
      (/
       (/ 1.0 F)
       (+
        (+
         (-
          (/ F (* PI l))
          (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (/ (pow PI 2.0) F)))
         (*
          (pow l 5.0)
          (*
           F
           (-
            (fma
             t_0
             -0.5
             (+
              (* (pow PI 5.0) 0.0125)
              (*
               0.3333333333333333
               (fma
                (pow PI 2.0)
                (-
                 (* (pow PI 3.0) 0.008333333333333333)
                 (* (pow PI 3.0) -0.013888888888888888))
                t_0))))
            (* -0.0001984126984126984 (pow PI 5.0))))))
        (*
         (pow l 3.0)
         (-
          (* (/ F (pow PI 3.0)) (pow (* (pow PI 3.0) 0.3333333333333333) 2.0))
          (*
           (/ F (pow PI 2.0))
           (-
            (* 0.008333333333333333 (pow PI 5.0))
            (fma
             -0.5
             (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 2.0)))
             (* (pow PI 5.0) 0.041666666666666664)))))))))
     (* PI l))))

C

double code(double F, double l) {
	double t_0 = pow(((double) M_PI), 5.0) * 0.13333333333333333;
	double tmp;
	if ((((double) M_PI) * l) <= 5e-17) {
		tmp = (((double) M_PI) * l) - ((1.0 / F) / ((((F / (((double) M_PI) * l)) - ((pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)) / (pow(((double) M_PI), 2.0) / F))) + (pow(l, 5.0) * (F * (fma(t_0, -0.5, ((pow(((double) M_PI), 5.0) * 0.0125) + (0.3333333333333333 * fma(pow(((double) M_PI), 2.0), ((pow(((double) M_PI), 3.0) * 0.008333333333333333) - (pow(((double) M_PI), 3.0) * -0.013888888888888888)), t_0)))) - (-0.0001984126984126984 * pow(((double) M_PI), 5.0)))))) + (pow(l, 3.0) * (((F / pow(((double) M_PI), 3.0)) * pow((pow(((double) M_PI), 3.0) * 0.3333333333333333), 2.0)) - ((F / pow(((double) M_PI), 2.0)) * ((0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 2.0))), (pow(((double) M_PI), 5.0) * 0.041666666666666664))))))));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Julia

function code(F, l)
	t_0 = Float64((pi ^ 5.0) * 0.13333333333333333)
	tmp = 0.0
	if (Float64(pi * l) <= 5e-17)
		tmp = Float64(Float64(pi * l) - Float64(Float64(1.0 / F) / Float64(Float64(Float64(Float64(F / Float64(pi * l)) - Float64(Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)) / Float64((pi ^ 2.0) / F))) + Float64((l ^ 5.0) * Float64(F * Float64(fma(t_0, -0.5, Float64(Float64((pi ^ 5.0) * 0.0125) + Float64(0.3333333333333333 * fma((pi ^ 2.0), Float64(Float64((pi ^ 3.0) * 0.008333333333333333) - Float64((pi ^ 3.0) * -0.013888888888888888)), t_0)))) - Float64(-0.0001984126984126984 * (pi ^ 5.0)))))) + Float64((l ^ 3.0) * Float64(Float64(Float64(F / (pi ^ 3.0)) * (Float64((pi ^ 3.0) * 0.3333333333333333) ^ 2.0)) - Float64(Float64(F / (pi ^ 2.0)) * Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 2.0))), Float64((pi ^ 5.0) * 0.041666666666666664)))))))));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]}, If[LessEqual[N[(Pi * l), $MachinePrecision], 5e-17], N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] / N[(N[(N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 5.0], $MachinePrecision] * N[(F * N[(N[(t$95$0 * -0.5 + N[(N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.0125), $MachinePrecision] + N[(0.3333333333333333 * N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.013888888888888888), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.0001984126984126984 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[(F / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(F / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.9999999999999999e-17

   1. Initial program 78.2%

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

      1. associate-/r/ 78.2%

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

      2. associate-/l* 86.9%

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

      3. clear-num 86.9%

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

      4. add-sqr-sqrt 42.1%

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

      5. sqrt-prod 63.1%

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

      6. sqr-neg 63.1%

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

      7. sqrt-unprod 26.2%

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

      8. add-sqr-sqrt 49.5%

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

      9. div-inv 49.5%

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

      10. metadata-eval 49.5%

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

      11. add-sqr-sqrt 23.3%

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

      12. sqrt-prod 64.5%

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

      13. sqrt-div 64.5%

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

      14. associate-*l/ 64.5%

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

      15. clear-num 64.5%

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

      16. associate-*l/ 64.6%

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

   3. Applied egg-rr 86.9%

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

   4. Taylor expanded in l around 0 96.1%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in F around 0 97.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\left(\frac{F}{\ell \cdot \pi} - \frac{{\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)}{\frac{{\pi}^{2}}{F}}\right) - \color{blue}{F \cdot \left({\ell}^{5} \cdot \left(-0.0001984126984126984 \cdot {\pi}^{5} - \left(-0.5 \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.16666666666666666 \cdot {\pi}^{5} + 0.041666666666666664 \cdot {\pi}^{5}\right)\right) + \left(-0.001388888888888889 \cdot {\pi}^{5} + \left(0.013888888888888888 \cdot {\pi}^{5} + \left(0.3333333333333333 \cdot \left({\pi}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - \left(-0.16666666666666666 \cdot {\pi}^{3} + \left(0.041666666666666664 \cdot {\pi}^{3} + 0.1111111111111111 \cdot {\pi}^{3}\right)\right)\right)\right) + 0.3333333333333333 \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.16666666666666666 \cdot {\pi}^{5} + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)\right)\right)\right)\right)\right)\right)}\right) - {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}\right)} \]

   9. Simplified 97.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\left(\frac{F}{\ell \cdot \pi} - \frac{{\pi}^{3} \cdot \left(0.3333333333333333 \cdot \ell\right)}{\frac{{\pi}^{2}}{F}}\right) - \color{blue}{{\ell}^{5} \cdot \left(F \cdot \left({\pi}^{5} \cdot -0.0001984126984126984 - \mathsf{fma}\left({\pi}^{5} \cdot 0.13333333333333333, -0.5, {\pi}^{5} \cdot 0.0125 + 0.3333333333333333 \cdot \mathsf{fma}\left({\pi}^{2}, 0.008333333333333333 \cdot {\pi}^{3} - {\pi}^{3} \cdot -0.013888888888888888, {\pi}^{5} \cdot 0.13333333333333333\right)\right)\right)\right)}\right) - {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}\right)} \]

   if 4.9999999999999999e-17 < (*.f64 (PI.f64) l)

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. fma-neg 64.1%

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

      5. associate-*l/ 64.1%

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

      6. times-frac 64.1%

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

      7. distribute-lft-neg-in 64.1%

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

      8. neg-mul-1 64.1%

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

      9. associate-/r* 64.1%

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

      10. metadata-eval 64.1%

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

      11. distribute-neg-frac 64.1%

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

      12. metadata-eval 64.1%

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

      13. times-frac 64.1%

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

   3. Simplified 64.1%

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

      1. fma-udef 64.1%

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

      2. associate-/l/ 64.1%

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

      3. associate-/r* 64.1%

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

      4. add-sqr-sqrt 37.4%

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

      5. sqrt-unprod 64.4%

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

      6. sqr-neg 64.4%

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

      7. sqrt-prod 26.9%

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

      8. add-sqr-sqrt 64.3%

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

      9. associate-/r* 64.3%

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

      10. div-inv 64.3%

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

      11. pow2 64.3%

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

      12. pow-flip 64.3%

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

      13. metadata-eval 64.3%

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

   5. Applied egg-rr 64.3%

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

      1. *-commutative 64.3%

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

      2. fma-def 64.3%

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

      3. *-commutative 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in l around inf 99.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\left(\left(\frac{F}{\pi \cdot \ell} - \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{\frac{{\pi}^{2}}{F}}\right) + {\ell}^{5} \cdot \left(F \cdot \left(\mathsf{fma}\left({\pi}^{5} \cdot 0.13333333333333333, -0.5, {\pi}^{5} \cdot 0.0125 + 0.3333333333333333 \cdot \mathsf{fma}\left({\pi}^{2}, {\pi}^{3} \cdot 0.008333333333333333 - {\pi}^{3} \cdot -0.013888888888888888, {\pi}^{5} \cdot 0.13333333333333333\right)\right) - -0.0001984126984126984 \cdot {\pi}^{5}\right)\right)\right) + {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} - \frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\pi \cdot \ell} - \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{\frac{{\pi}^{2}}{F}}\right) + {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} - \frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 5e-17)
   (-
    (* PI l)
    (/
     (/ 1.0 F)
     (+
      (-
       (/ F (* PI l))
       (/ (* (pow PI 3.0) (* l 0.3333333333333333)) (/ (pow PI 2.0) F)))
      (*
       (pow l 3.0)
       (-
        (* (/ F (pow PI 3.0)) (pow (* (pow PI 3.0) 0.3333333333333333) 2.0))
        (*
         (/ F (pow PI 2.0))
         (-
          (* 0.008333333333333333 (pow PI 5.0))
          (fma
           -0.5
           (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 2.0)))
           (* (pow PI 5.0) 0.041666666666666664)))))))))
   (* PI l)))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 5e-17) {
		tmp = (((double) M_PI) * l) - ((1.0 / F) / (((F / (((double) M_PI) * l)) - ((pow(((double) M_PI), 3.0) * (l * 0.3333333333333333)) / (pow(((double) M_PI), 2.0) / F))) + (pow(l, 3.0) * (((F / pow(((double) M_PI), 3.0)) * pow((pow(((double) M_PI), 3.0) * 0.3333333333333333), 2.0)) - ((F / pow(((double) M_PI), 2.0)) * ((0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 2.0))), (pow(((double) M_PI), 5.0) * 0.041666666666666664))))))));
	} else {
		tmp = ((double) M_PI) * l;
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= 5e-17)
		tmp = Float64(Float64(pi * l) - Float64(Float64(1.0 / F) / Float64(Float64(Float64(F / Float64(pi * l)) - Float64(Float64((pi ^ 3.0) * Float64(l * 0.3333333333333333)) / Float64((pi ^ 2.0) / F))) + Float64((l ^ 3.0) * Float64(Float64(Float64(F / (pi ^ 3.0)) * (Float64((pi ^ 3.0) * 0.3333333333333333) ^ 2.0)) - Float64(Float64(F / (pi ^ 2.0)) * Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 2.0))), Float64((pi ^ 5.0) * 0.041666666666666664)))))))));
	else
		tmp = Float64(pi * l);
	end
	return tmp
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 5e-17], N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] / N[(N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[(F / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(F / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.9999999999999999e-17

   1. Initial program 78.2%

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

      1. associate-/r/ 78.2%

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

      2. associate-/l* 86.9%

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

      3. clear-num 86.9%

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

      4. add-sqr-sqrt 42.1%

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

      5. sqrt-prod 63.1%

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

      6. sqr-neg 63.1%

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

      7. sqrt-unprod 26.2%

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

      8. add-sqr-sqrt 49.5%

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

      9. div-inv 49.5%

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

      10. metadata-eval 49.5%

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

      11. add-sqr-sqrt 23.3%

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

      12. sqrt-prod 64.5%

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

      13. sqrt-div 64.5%

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

      14. associate-*l/ 64.5%

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

      15. clear-num 64.5%

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

      16. associate-*l/ 64.6%

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

   3. Applied egg-rr 86.9%

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

   4. Taylor expanded in l around 0 96.6%

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

   6. Simplified 97.6%

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

   if 4.9999999999999999e-17 < (*.f64 (PI.f64) l)

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. fma-neg 64.1%

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

      5. associate-*l/ 64.1%

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

      6. times-frac 64.1%

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

      7. distribute-lft-neg-in 64.1%

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

      8. neg-mul-1 64.1%

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

      9. associate-/r* 64.1%

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

      10. metadata-eval 64.1%

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

      11. distribute-neg-frac 64.1%

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

      12. metadata-eval 64.1%

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

      13. times-frac 64.1%

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

   3. Simplified 64.1%

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

      1. fma-udef 64.1%

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

      2. associate-/l/ 64.1%

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

      3. associate-/r* 64.1%

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

      4. add-sqr-sqrt 37.4%

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

      5. sqrt-unprod 64.4%

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

      6. sqr-neg 64.4%

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

      7. sqrt-prod 26.9%

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

      8. add-sqr-sqrt 64.3%

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

      9. associate-/r* 64.3%

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

      10. div-inv 64.3%

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

      11. pow2 64.3%

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

      12. pow-flip 64.3%

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

      13. metadata-eval 64.3%

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

   5. Applied egg-rr 64.3%

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

      1. *-commutative 64.3%

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

      2. fma-def 64.3%

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

      3. *-commutative 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in l around inf 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\pi \cdot \ell} - \frac{{\pi}^{3} \cdot \left(\ell \cdot 0.3333333333333333\right)}{\frac{{\pi}^{2}}{F}}\right) + {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{3}} \cdot {\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2} - \frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+25) (not (<= (* PI l) 5e-17)))
   (* PI l)
   (- (* PI l) (* (/ PI F) (/ l F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+25) || !((((double) M_PI) * l) <= 5e-17)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) * (l / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+25) || !((Math.PI * l) <= 5e-17)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) * (l / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+25) or not ((math.pi * l) <= 5e-17):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi / F) * (l / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+25) || !(Float64(pi * l) <= 5e-17))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) * Float64(l / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+25) || ~(((pi * l) <= 5e-17)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi / F) * (l / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+25], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-17]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2.00000000000000018e25 or 4.9999999999999999e-17 < (*.f64 (PI.f64) l)

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

      2. sqr-neg 59.0%

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

      3. *-commutative 59.0%

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

      4. fma-neg 59.0%

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

      5. associate-*l/ 59.0%

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

      6. times-frac 59.0%

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

      7. distribute-lft-neg-in 59.0%

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

      8. neg-mul-1 59.0%

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

      9. associate-/r* 59.0%

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

      10. metadata-eval 59.0%

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

      11. distribute-neg-frac 59.0%

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

      12. metadata-eval 59.0%

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

      13. times-frac 59.0%

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

   3. Simplified 59.0%

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

      1. fma-udef 59.0%

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

      2. associate-/l/ 59.0%

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

      3. associate-/r* 59.0%

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

      4. add-sqr-sqrt 35.4%

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

      5. sqrt-unprod 59.1%

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

      6. sqr-neg 59.1%

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

      7. sqrt-prod 23.7%

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

      8. add-sqr-sqrt 59.0%

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

      9. associate-/r* 59.0%

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

      10. div-inv 59.0%

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

      11. pow2 59.0%

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

      12. pow-flip 59.0%

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

      13. metadata-eval 59.0%

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

   5. Applied egg-rr 59.0%

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

      1. *-commutative 59.0%

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

      2. fma-def 59.0%

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

      3. *-commutative 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in l around inf 99.7%

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

   if -2.00000000000000018e25 < (*.f64 (PI.f64) l) < 4.9999999999999999e-17

   1. Initial program 87.6%

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

      1. sqr-neg 87.6%

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

      2. associate-*l/ 87.7%

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

      3. *-lft-identity 87.7%

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

      4. sqr-neg 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in l around 0 86.2%

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

      1. *-commutative 86.2%

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

      2. times-frac 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+25} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 6: 74.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell \end{array} \]

FPCore

(FPCore (F l) :precision binary64 (* PI l))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[F_, l_] := N[(Pi * l), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

   1. *-commutative 74.6%

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

   2. sqr-neg 74.6%

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

   3. *-commutative 74.6%

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

   4. fma-neg 74.6%

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

   5. associate-*l/ 74.7%

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

   6. times-frac 81.2%

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

   7. distribute-lft-neg-in 81.2%

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

   8. neg-mul-1 81.2%

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

   9. associate-/r* 81.2%

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

   10. metadata-eval 81.2%

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

   11. distribute-neg-frac 81.2%

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

   12. metadata-eval 81.2%

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

   13. times-frac 74.7%

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

3. Simplified 81.2%

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

   1. fma-udef 81.2%

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

   2. associate-/l/ 74.7%

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

   3. associate-/r* 81.2%

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

   4. add-sqr-sqrt 42.8%

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

   5. sqrt-unprod 64.5%

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

   6. sqr-neg 64.5%

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

   7. sqrt-prod 24.2%

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

   8. add-sqr-sqrt 53.2%

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

   9. associate-/r* 53.2%

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

   10. div-inv 53.2%

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

   11. pow2 53.2%

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

   12. pow-flip 53.2%

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

   13. metadata-eval 53.2%

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

5. Applied egg-rr 53.2%

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

   1. *-commutative 53.2%

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

   2. fma-def 53.2%

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

   3. *-commutative 53.2%

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

7. Simplified 53.2%

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

8. Taylor expanded in l around inf 71.9%

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

9. Final simplification 71.9%

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

Reproduce

herbie shell --seed 2023299 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))