VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.0% → 88.3%

Time: 33.5s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.1993518130896656e-156
  2. l = 3754087061682184.5

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: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{F}^{2}}{{\pi}^{2}}\\ t_1 := 0.008333333333333333 \cdot {\pi}^{5}\\ t_2 := \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\\ t_3 := t\_1 - t\_2\\ t_4 := {\pi}^{3} \cdot 0.3333333333333333\\ t_5 := {F}^{2} \cdot \frac{{t\_4}^{2}}{{\pi}^{3}}\\ \mathbf{if}\;F \cdot F \leq 10^{-322} \lor \neg \left(F \cdot F \leq 10^{-88}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, {\ell}^{2} \cdot \left(t\_4 \cdot \frac{t\_3 \cdot t\_0 - t\_5}{\pi} + \left(\left(t\_3 \cdot t\_4\right) \cdot \frac{{F}^{2}}{{\pi}^{3}} + t\_0 \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{2} \cdot t\_3, \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right) + \left(t\_5 + t\_0 \cdot \left(t\_2 - t\_1\right)\right), {F}^{2} \cdot \left({\pi}^{3} \cdot \frac{0.3333333333333333}{-{\pi}^{2}}\right)\right), \frac{{F}^{2}}{\pi}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (/ (pow F 2.0) (pow PI 2.0)))
        (t_1 (* 0.008333333333333333 (pow PI 5.0)))
        (t_2
         (fma
          -0.5
          (* (* (pow PI 2.0) (pow PI 3.0)) 0.3333333333333333)
          (* (pow PI 5.0) 0.041666666666666664)))
        (t_3 (- t_1 t_2))
        (t_4 (* (pow PI 3.0) 0.3333333333333333))
        (t_5 (* (pow F 2.0) (/ (pow t_4 2.0) (pow PI 3.0)))))
   (if (or (<= (* F F) 1e-322) (not (<= (* F F) 1e-88)))
     (- (* PI l) (/ (/ (tan (* PI l)) F) F))
     (-
      (* PI l)
      (/
       1.0
       (/
        (fma
         (pow l 2.0)
         (fma
          (pow l 2.0)
          (+
           (*
            (pow l 2.0)
            (+
             (* t_4 (/ (- (* t_3 t_0) t_5) PI))
             (+
              (* (* t_3 t_4) (/ (pow F 2.0) (pow PI 3.0)))
              (*
               t_0
               (-
                (fma
                 -0.5
                 (* (pow PI 2.0) t_3)
                 (fma
                  0.041666666666666664
                  (* (pow PI 3.0) (* 0.3333333333333333 (pow PI 4.0)))
                  (* (pow PI 7.0) -0.001388888888888889)))
                (* -0.0001984126984126984 (pow PI 7.0)))))))
           (+ t_5 (* t_0 (- t_2 t_1))))
          (*
           (pow F 2.0)
           (* (pow PI 3.0) (/ 0.3333333333333333 (- (pow PI 2.0))))))
         (/ (pow F 2.0) PI))
        l))))))

C

double code(double F, double l) {
	double t_0 = pow(F, 2.0) / pow(((double) M_PI), 2.0);
	double t_1 = 0.008333333333333333 * pow(((double) M_PI), 5.0);
	double t_2 = fma(-0.5, ((pow(((double) M_PI), 2.0) * pow(((double) M_PI), 3.0)) * 0.3333333333333333), (pow(((double) M_PI), 5.0) * 0.041666666666666664));
	double t_3 = t_1 - t_2;
	double t_4 = pow(((double) M_PI), 3.0) * 0.3333333333333333;
	double t_5 = pow(F, 2.0) * (pow(t_4, 2.0) / pow(((double) M_PI), 3.0));
	double tmp;
	if (((F * F) <= 1e-322) || !((F * F) <= 1e-88)) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) - (1.0 / (fma(pow(l, 2.0), fma(pow(l, 2.0), ((pow(l, 2.0) * ((t_4 * (((t_3 * t_0) - t_5) / ((double) M_PI))) + (((t_3 * t_4) * (pow(F, 2.0) / pow(((double) M_PI), 3.0))) + (t_0 * (fma(-0.5, (pow(((double) M_PI), 2.0) * t_3), fma(0.041666666666666664, (pow(((double) M_PI), 3.0) * (0.3333333333333333 * pow(((double) M_PI), 4.0))), (pow(((double) M_PI), 7.0) * -0.001388888888888889))) - (-0.0001984126984126984 * pow(((double) M_PI), 7.0))))))) + (t_5 + (t_0 * (t_2 - t_1)))), (pow(F, 2.0) * (pow(((double) M_PI), 3.0) * (0.3333333333333333 / -pow(((double) M_PI), 2.0))))), (pow(F, 2.0) / ((double) M_PI))) / l));
	}
	return tmp;
}

Julia

function code(F, l)
	t_0 = Float64((F ^ 2.0) / (pi ^ 2.0))
	t_1 = Float64(0.008333333333333333 * (pi ^ 5.0))
	t_2 = fma(-0.5, Float64(Float64((pi ^ 2.0) * (pi ^ 3.0)) * 0.3333333333333333), Float64((pi ^ 5.0) * 0.041666666666666664))
	t_3 = Float64(t_1 - t_2)
	t_4 = Float64((pi ^ 3.0) * 0.3333333333333333)
	t_5 = Float64((F ^ 2.0) * Float64((t_4 ^ 2.0) / (pi ^ 3.0)))
	tmp = 0.0
	if ((Float64(F * F) <= 1e-322) || !(Float64(F * F) <= 1e-88))
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	else
		tmp = Float64(Float64(pi * l) - Float64(1.0 / Float64(fma((l ^ 2.0), fma((l ^ 2.0), Float64(Float64((l ^ 2.0) * Float64(Float64(t_4 * Float64(Float64(Float64(t_3 * t_0) - t_5) / pi)) + Float64(Float64(Float64(t_3 * t_4) * Float64((F ^ 2.0) / (pi ^ 3.0))) + Float64(t_0 * Float64(fma(-0.5, Float64((pi ^ 2.0) * t_3), fma(0.041666666666666664, Float64((pi ^ 3.0) * Float64(0.3333333333333333 * (pi ^ 4.0))), Float64((pi ^ 7.0) * -0.001388888888888889))) - Float64(-0.0001984126984126984 * (pi ^ 7.0))))))) + Float64(t_5 + Float64(t_0 * Float64(t_2 - t_1)))), Float64((F ^ 2.0) * Float64((pi ^ 3.0) * Float64(0.3333333333333333 / Float64(-(pi ^ 2.0)))))), Float64((F ^ 2.0) / pi)) / l)));
	end
	return tmp
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(F * F), $MachinePrecision], 1e-322], N[Not[LessEqual[N[(F * F), $MachinePrecision], 1e-88]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(t$95$4 * N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] - t$95$5), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$3 * t$95$4), $MachinePrecision] * N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(-0.5 * N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$3), $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] - N[(-0.0001984126984126984 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + N[(t$95$0 * N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 / (-N[Power[Pi, 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 9.88131e-323 or 9.99999999999999934e-89 < (*.f64 F F)

   1. Initial program 81.1%

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

      1. associate-*l/ 81.3%

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

      2. *-un-lft-identity 81.3%

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

      3. associate-/r* 89.3%

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

   4. Applied egg-rr 89.3%

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

   if 9.88131e-323 < (*.f64 F F) < 9.99999999999999934e-89

   1. Initial program 68.6%

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

      1. associate-/r/ 68.7%

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

      2. pow2 68.7%

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

   4. Applied egg-rr 68.7%

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

   5. Taylor expanded in l around 0 95.6%

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

   7. Simplified 95.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-322} \lor \neg \left(F \cdot F \leq 10^{-88}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, {\ell}^{2} \cdot \left(\left({\pi}^{3} \cdot 0.3333333333333333\right) \cdot \frac{\left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{2}} - {F}^{2} \cdot \frac{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}{{\pi}^{3}}}{\pi} + \left(\left(\left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{3}} + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left(-0.5, {\pi}^{2} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right), \mathsf{fma}\left(0.041666666666666664, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{4}\right), {\pi}^{7} \cdot -0.001388888888888889\right)\right) - -0.0001984126984126984 \cdot {\pi}^{7}\right)\right)\right) + \left({F}^{2} \cdot \frac{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}{{\pi}^{3}} + \frac{{F}^{2}}{{\pi}^{2}} \cdot \left(\mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right) - 0.008333333333333333 \cdot {\pi}^{5}\right)\right), {F}^{2} \cdot \left({\pi}^{3} \cdot \frac{0.3333333333333333}{-{\pi}^{2}}\right)\right), \frac{{F}^{2}}{\pi}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-322} \lor \neg \left(F \cdot F \leq 10^{-88}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(\left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{2}} - {F}^{2} \cdot \frac{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}{{\pi}^{3}}, -{\ell}^{2}, {F}^{2} \cdot \left({\pi}^{3} \cdot \frac{0.3333333333333333}{-{\pi}^{2}}\right)\right), \frac{{F}^{2}}{\pi}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* F F) 1e-322) (not (<= (* F F) 1e-88)))
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))
   (-
    (* PI l)
    (/
     1.0
     (/
      (fma
       (pow l 2.0)
       (fma
        (-
         (*
          (-
           (* 0.008333333333333333 (pow PI 5.0))
           (fma
            -0.5
            (* (* (pow PI 2.0) (pow PI 3.0)) 0.3333333333333333)
            (* (pow PI 5.0) 0.041666666666666664)))
          (/ (pow F 2.0) (pow PI 2.0)))
         (*
          (pow F 2.0)
          (/ (pow (* (pow PI 3.0) 0.3333333333333333) 2.0) (pow PI 3.0))))
        (- (pow l 2.0))
        (*
         (pow F 2.0)
         (* (pow PI 3.0) (/ 0.3333333333333333 (- (pow PI 2.0))))))
       (/ (pow F 2.0) PI))
      l)))))

C

double code(double F, double l) {
	double tmp;
	if (((F * F) <= 1e-322) || !((F * F) <= 1e-88)) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) - (1.0 / (fma(pow(l, 2.0), fma(((((0.008333333333333333 * pow(((double) M_PI), 5.0)) - fma(-0.5, ((pow(((double) M_PI), 2.0) * pow(((double) M_PI), 3.0)) * 0.3333333333333333), (pow(((double) M_PI), 5.0) * 0.041666666666666664))) * (pow(F, 2.0) / pow(((double) M_PI), 2.0))) - (pow(F, 2.0) * (pow((pow(((double) M_PI), 3.0) * 0.3333333333333333), 2.0) / pow(((double) M_PI), 3.0)))), -pow(l, 2.0), (pow(F, 2.0) * (pow(((double) M_PI), 3.0) * (0.3333333333333333 / -pow(((double) M_PI), 2.0))))), (pow(F, 2.0) / ((double) M_PI))) / l));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(F * F) <= 1e-322) || !(Float64(F * F) <= 1e-88))
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	else
		tmp = Float64(Float64(pi * l) - Float64(1.0 / Float64(fma((l ^ 2.0), fma(Float64(Float64(Float64(Float64(0.008333333333333333 * (pi ^ 5.0)) - fma(-0.5, Float64(Float64((pi ^ 2.0) * (pi ^ 3.0)) * 0.3333333333333333), Float64((pi ^ 5.0) * 0.041666666666666664))) * Float64((F ^ 2.0) / (pi ^ 2.0))) - Float64((F ^ 2.0) * Float64((Float64((pi ^ 3.0) * 0.3333333333333333) ^ 2.0) / (pi ^ 3.0)))), Float64(-(l ^ 2.0)), Float64((F ^ 2.0) * Float64((pi ^ 3.0) * Float64(0.3333333333333333 / Float64(-(pi ^ 2.0)))))), Float64((F ^ 2.0) / pi)) / l)));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(F * F), $MachinePrecision], 1e-322], N[Not[LessEqual[N[(F * F), $MachinePrecision], 1e-88]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[(N[(N[(0.008333333333333333 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[F, 2.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Power[l, 2.0], $MachinePrecision]) + N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 / (-N[Power[Pi, 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[F, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 9.88131e-323 or 9.99999999999999934e-89 < (*.f64 F F)

   1. Initial program 81.1%

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

      1. associate-*l/ 81.3%

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

      2. *-un-lft-identity 81.3%

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

      3. associate-/r* 89.3%

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

   4. Applied egg-rr 89.3%

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

   if 9.88131e-323 < (*.f64 F F) < 9.99999999999999934e-89

   1. Initial program 68.6%

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

      1. associate-/r/ 68.7%

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

      2. pow2 68.7%

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

   4. Applied egg-rr 68.7%

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

   5. Taylor expanded in l around 0 93.8%

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

   7. Simplified 93.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-322} \lor \neg \left(F \cdot F \leq 10^{-88}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(\left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, \left({\pi}^{2} \cdot {\pi}^{3}\right) \cdot 0.3333333333333333, {\pi}^{5} \cdot 0.041666666666666664\right)\right) \cdot \frac{{F}^{2}}{{\pi}^{2}} - {F}^{2} \cdot \frac{{\left({\pi}^{3} \cdot 0.3333333333333333\right)}^{2}}{{\pi}^{3}}, -{\ell}^{2}, {F}^{2} \cdot \left({\pi}^{3} \cdot \frac{0.3333333333333333}{-{\pi}^{2}}\right)\right), \frac{{F}^{2}}{\pi}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 2e-69)
   (- (* PI l) (/ (/ (* PI l) F) F))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 2e-69) {
		tmp = (((double) M_PI) * l) - (((((double) M_PI) * l) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((Math.PI * l) <= 2e-69) {
		tmp = (Math.PI * l) - (((Math.PI * l) / F) / F);
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (math.pi * l) <= 2e-69:
		tmp = (math.pi * l) - (((math.pi * l) / F) / F)
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= 2e-69)
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / F));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((pi * l) <= 2e-69)
		tmp = (pi * l) - (((pi * l) / F) / F);
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 2e-69], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.9999999999999999e-69

   1. Initial program 80.3%

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

      1. associate-*l/ 80.4%

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

      2. *-un-lft-identity 80.4%

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

      3. associate-/r* 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in l around 0 84.8%

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

   if 1.9999999999999999e-69 < (*.f64 (PI.f64) l)

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

      2. sqr-neg 74.5%

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

      3. associate-*r/ 74.5%

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

      4. sqr-neg 74.5%

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

      5. *-rgt-identity 74.5%

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

   3. Simplified 74.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 78.7%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. associate-/r* 85.3%

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

4. Applied egg-rr 85.3%

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

5. Final simplification 85.3%

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

Alternative 5: 75.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. *-commutative 78.7%

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

   2. sqr-neg 78.7%

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

   3. associate-*r/ 78.8%

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

   4. sqr-neg 78.8%

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

   5. *-rgt-identity 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in l around 0 71.9%

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

   1. *-commutative 71.9%

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

   2. times-frac 78.2%

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

7. Applied egg-rr 78.2%

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

8. Final simplification 78.2%

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

Alternative 6: 75.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. *-commutative 78.7%

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

   2. sqr-neg 78.7%

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

   3. associate-*r/ 78.8%

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

   4. sqr-neg 78.8%

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

   5. *-rgt-identity 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in l around 0 71.9%

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

   1. *-commutative 71.9%

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

   2. times-frac 78.2%

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

7. Applied egg-rr 78.2%

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

   1. *-commutative 78.2%

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

   2. clear-num 78.2%

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

   3. frac-times 78.3%

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

   4. *-un-lft-identity 78.3%

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

9. Applied egg-rr 78.3%

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

10. Final simplification 78.3%

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

Alternative 7: 75.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. associate-/r* 85.3%

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

4. Applied egg-rr 85.3%

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

5. Taylor expanded in l around 0 78.3%

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

6. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2024067 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))