VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 99.2%

Time: 1.0min

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. F = 7.123829694300288e+132
  2. l = 4105893128605720.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{t\_0 \cdot \frac{1}{\cos \left(l\_m \cdot \sqrt[3]{{\pi}^{3}}\right)}}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{t\_0 \cdot \frac{1}{1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(-0.001388888888888889 \cdot \left({l\_m}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (sin (* PI l_m))))
   (*
    l_s
    (if (<= (* PI l_m) 2e+14)
      (+
       (* PI l_m)
       (/ -1.0 (/ F (/ (* t_0 (/ 1.0 (cos (* l_m (cbrt (pow PI 3.0)))))) F))))
      (+
       (* PI l_m)
       (/
        -1.0
        (/
         F
         (/
          (*
           t_0
           (/
            1.0
            (+
             1.0
             (*
              (pow l_m 2.0)
              (+
               (* -0.5 (pow PI 2.0))
               (*
                (pow l_m 2.0)
                (+
                 (*
                  -0.001388888888888889
                  (* (pow l_m 2.0) (log1p (expm1 (pow PI 6.0)))))
                 (* 0.041666666666666664 (pow PI 4.0)))))))))
          F))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = sin((((double) M_PI) * l_m));
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e+14) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F / ((t_0 * (1.0 / cos((l_m * cbrt(pow(((double) M_PI), 3.0)))))) / F)));
	} else {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F / ((t_0 * (1.0 / (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((-0.001388888888888889 * (pow(l_m, 2.0) * log1p(expm1(pow(((double) M_PI), 6.0))))) + (0.041666666666666664 * pow(((double) M_PI), 4.0))))))))) / F)));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = Math.sin((Math.PI * l_m));
	double tmp;
	if ((Math.PI * l_m) <= 2e+14) {
		tmp = (Math.PI * l_m) + (-1.0 / (F / ((t_0 * (1.0 / Math.cos((l_m * Math.cbrt(Math.pow(Math.PI, 3.0)))))) / F)));
	} else {
		tmp = (Math.PI * l_m) + (-1.0 / (F / ((t_0 * (1.0 / (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.log1p(Math.expm1(Math.pow(Math.PI, 6.0))))) + (0.041666666666666664 * Math.pow(Math.PI, 4.0))))))))) / F)));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = sin(Float64(pi * l_m))
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e+14)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(t_0 * Float64(1.0 / cos(Float64(l_m * cbrt((pi ^ 3.0)))))) / F))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * log1p(expm1((pi ^ 6.0))))) + Float64(0.041666666666666664 * (pi ^ 4.0))))))))) / F))));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e+14], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(t$95$0 * N[(1.0 / N[Cos[N[(l$95$m * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(t$95$0 * N[(1.0 / N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Log[1 + N[(Exp[N[Power[Pi, 6.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 82.1%

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

      1. associate-*l/ 82.2%

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

      2. *-un-lft-identity 82.2%

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

      3. associate-/r* 87.9%

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

      4. clear-num 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. tan-quot 87.9%

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

      2. div-inv 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. add-cbrt-cube 87.9%

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

      2. pow3 87.9%

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

   8. Applied egg-rr 87.9%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 57.4%

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

      1. associate-*l/ 57.4%

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

      2. *-un-lft-identity 57.4%

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

      3. associate-/r* 57.4%

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

      4. clear-num 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. tan-quot 57.4%

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

      2. div-inv 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in l around 0 93.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F}}} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 99.6%

         \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\left|\sin \left(\pi \cdot l\_m\right)\right| \cdot \frac{1}{1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({l\_m}^{2} \cdot {\pi}^{6}\right)\right)\right)}}{F}}}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    -1.0
    (/
     F
     (/
      (*
       (fabs (sin (* PI l_m)))
       (/
        1.0
        (+
         1.0
         (*
          (pow l_m 2.0)
          (+
           (* -0.5 (pow PI 2.0))
           (*
            (pow l_m 2.0)
            (+
             (* 0.041666666666666664 (pow PI 4.0))
             (* -0.001388888888888889 (* (pow l_m 2.0) (pow PI 6.0))))))))))
      F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (-1.0 / (F / ((fabs(sin((((double) M_PI) * l_m))) * (1.0 / (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((0.041666666666666664 * pow(((double) M_PI), 4.0)) + (-0.001388888888888889 * (pow(l_m, 2.0) * pow(((double) M_PI), 6.0)))))))))) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (-1.0 / (F / ((Math.abs(Math.sin((Math.PI * l_m))) * (1.0 / (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((0.041666666666666664 * Math.pow(Math.PI, 4.0)) + (-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 6.0)))))))))) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (-1.0 / (F / ((math.fabs(math.sin((math.pi * l_m))) * (1.0 / (1.0 + (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((0.041666666666666664 * math.pow(math.pi, 4.0)) + (-0.001388888888888889 * (math.pow(l_m, 2.0) * math.pow(math.pi, 6.0)))))))))) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(abs(sin(Float64(pi * l_m))) * Float64(1.0 / Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(0.041666666666666664 * (pi ^ 4.0)) + Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * (pi ^ 6.0)))))))))) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (-1.0 / (F / ((abs(sin((pi * l_m))) * (1.0 / (1.0 + ((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l_m ^ 2.0) * ((0.041666666666666664 * (pi ^ 4.0)) + (-0.001388888888888889 * ((l_m ^ 2.0) * (pi ^ 6.0)))))))))) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(N[Abs[N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

   1. tan-quot 80.2%

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

   2. div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in l around 0 96.2%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F}}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 44.8%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\left(\sqrt{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt{\sin \left(\pi \cdot \ell\right)}\right)} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

   2. sqrt-unprod 80.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\sqrt{\sin \left(\pi \cdot \ell\right) \cdot \sin \left(\pi \cdot \ell\right)}} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

   3. pow2 80.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sqrt{\color{blue}{{\sin \left(\pi \cdot \ell\right)}^{2}}} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

9. Applied egg-rr 80.6%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\sqrt{{\sin \left(\pi \cdot \ell\right)}^{2}}} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]
10. Step-by-step derivation

    1. unpow2 80.6%

       \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sqrt{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \sin \left(\pi \cdot \ell\right)}} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

    2. rem-sqrt-square 84.2%

       \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\left|\sin \left(\pi \cdot \ell\right)\right|} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

    3. *-commutative 84.2%

       \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\left|\sin \color{blue}{\left(\ell \cdot \pi\right)}\right| \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

11. Simplified 84.2%

    \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\left|\sin \left(\ell \cdot \pi\right)\right|} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

12. Final simplification 84.2%

    \[\leadsto \pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\left|\sin \left(\pi \cdot \ell\right)\right| \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right)\right)\right)}}{F}}} \]
13. Add Preprocessing

Alternative 3: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{1}{1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\pi}^{6} \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right)}}{F}}}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    -1.0
    (/
     F
     (/
      (*
       (sin (* PI l_m))
       (/
        1.0
        (+
         1.0
         (*
          (pow l_m 2.0)
          (+
           (* -0.5 (pow PI 2.0))
           (*
            (pow l_m 2.0)
            (+
             (* 0.041666666666666664 (pow PI 4.0))
             (* -0.001388888888888889 (* (pow PI 6.0) (* l_m l_m))))))))))
      F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (-1.0 / (F / ((sin((((double) M_PI) * l_m)) * (1.0 / (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((0.041666666666666664 * pow(((double) M_PI), 4.0)) + (-0.001388888888888889 * (pow(((double) M_PI), 6.0) * (l_m * l_m)))))))))) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (-1.0 / (F / ((Math.sin((Math.PI * l_m)) * (1.0 / (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((0.041666666666666664 * Math.pow(Math.PI, 4.0)) + (-0.001388888888888889 * (Math.pow(Math.PI, 6.0) * (l_m * l_m)))))))))) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (-1.0 / (F / ((math.sin((math.pi * l_m)) * (1.0 / (1.0 + (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((0.041666666666666664 * math.pow(math.pi, 4.0)) + (-0.001388888888888889 * (math.pow(math.pi, 6.0) * (l_m * l_m)))))))))) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(sin(Float64(pi * l_m)) * Float64(1.0 / Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(0.041666666666666664 * (pi ^ 4.0)) + Float64(-0.001388888888888889 * Float64((pi ^ 6.0) * Float64(l_m * l_m)))))))))) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (-1.0 / (F / ((sin((pi * l_m)) * (1.0 / (1.0 + ((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l_m ^ 2.0) * ((0.041666666666666664 * (pi ^ 4.0)) + (-0.001388888888888889 * ((pi ^ 6.0) * (l_m * l_m)))))))))) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.001388888888888889 * N[(N[Power[Pi, 6.0], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

   1. tan-quot 80.2%

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

   2. div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in l around 0 96.2%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F}}} \]
8. Step-by-step derivation

   1. unpow2 96.2%

      \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

9. Applied egg-rr 96.2%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

10. Final simplification 96.2%

    \[\leadsto \pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\pi}^{6} \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}}{F}}} \]
11. Add Preprocessing

Alternative 4: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{1}{\frac{F}{\frac{\left(\pi \cdot l\_m\right) \cdot \frac{1}{-1 - {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({l\_m}^{2} \cdot {\pi}^{6}\right)\right)\right)}}{F}}}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    1.0
    (/
     F
     (/
      (*
       (* PI l_m)
       (/
        1.0
        (-
         -1.0
         (*
          (pow l_m 2.0)
          (+
           (* -0.5 (pow PI 2.0))
           (*
            (pow l_m 2.0)
            (+
             (* 0.041666666666666664 (pow PI 4.0))
             (* -0.001388888888888889 (* (pow l_m 2.0) (pow PI 6.0))))))))))
      F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (1.0 / (F / (((((double) M_PI) * l_m) * (1.0 / (-1.0 - (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((0.041666666666666664 * pow(((double) M_PI), 4.0)) + (-0.001388888888888889 * (pow(l_m, 2.0) * pow(((double) M_PI), 6.0)))))))))) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (1.0 / (F / (((Math.PI * l_m) * (1.0 / (-1.0 - (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((0.041666666666666664 * Math.pow(Math.PI, 4.0)) + (-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 6.0)))))))))) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (1.0 / (F / (((math.pi * l_m) * (1.0 / (-1.0 - (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((0.041666666666666664 * math.pow(math.pi, 4.0)) + (-0.001388888888888889 * (math.pow(l_m, 2.0) * math.pow(math.pi, 6.0)))))))))) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(1.0 / Float64(F / Float64(Float64(Float64(pi * l_m) * Float64(1.0 / Float64(-1.0 - Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(0.041666666666666664 * (pi ^ 4.0)) + Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * (pi ^ 6.0)))))))))) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (1.0 / (F / (((pi * l_m) * (1.0 / (-1.0 - ((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l_m ^ 2.0) * ((0.041666666666666664 * (pi ^ 4.0)) + (-0.001388888888888889 * ((l_m ^ 2.0) * (pi ^ 6.0)))))))))) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(1.0 / N[(F / N[(N[(N[(Pi * l$95$m), $MachinePrecision] * N[(1.0 / N[(-1.0 - N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

   1. tan-quot 80.2%

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

   2. div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in l around 0 96.2%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F}}} \]

8. Taylor expanded in l around 0 95.9%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\color{blue}{\left(\ell \cdot \pi\right)} \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}}} \]

9. Final simplification 95.9%

   \[\leadsto \pi \cdot \ell + \frac{1}{\frac{F}{\frac{\left(\pi \cdot \ell\right) \cdot \frac{1}{-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right)\right)\right)}}{F}}} \]
10. Add Preprocessing

Alternative 5: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{1}{1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({l\_m}^{2} \cdot {\pi}^{4}\right)\right)}}{F}}}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    -1.0
    (/
     F
     (/
      (*
       (sin (* PI l_m))
       (/
        1.0
        (+
         1.0
         (*
          (pow l_m 2.0)
          (+
           (* -0.5 (pow PI 2.0))
           (* 0.041666666666666664 (* (pow l_m 2.0) (pow PI 4.0))))))))
      F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (-1.0 / (F / ((sin((((double) M_PI) * l_m)) * (1.0 / (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (0.041666666666666664 * (pow(l_m, 2.0) * pow(((double) M_PI), 4.0)))))))) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (-1.0 / (F / ((Math.sin((Math.PI * l_m)) * (1.0 / (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (0.041666666666666664 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 4.0)))))))) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (-1.0 / (F / ((math.sin((math.pi * l_m)) * (1.0 / (1.0 + (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (0.041666666666666664 * (math.pow(l_m, 2.0) * math.pow(math.pi, 4.0)))))))) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(sin(Float64(pi * l_m)) * Float64(1.0 / Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64(0.041666666666666664 * Float64((l_m ^ 2.0) * (pi ^ 4.0)))))))) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (-1.0 / (F / ((sin((pi * l_m)) * (1.0 / (1.0 + ((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + (0.041666666666666664 * ((l_m ^ 2.0) * (pi ^ 4.0)))))))) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

   1. tan-quot 80.2%

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

   2. div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in l around 0 94.3%

   \[\leadsto \pi \cdot \ell - \frac{1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}}{F}}} \]

8. Final simplification 94.3%

   \[\leadsto \pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}{F}}} \]
9. Add Preprocessing

Alternative 6: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{1}{1 + -0.5 \cdot {\left(\pi \cdot l\_m\right)}^{2}}}{F}}}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    -1.0
    (/
     F
     (/
      (* (sin (* PI l_m)) (/ 1.0 (+ 1.0 (* -0.5 (pow (* PI l_m) 2.0)))))
      F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (-1.0 / (F / ((sin((((double) M_PI) * l_m)) * (1.0 / (1.0 + (-0.5 * pow((((double) M_PI) * l_m), 2.0))))) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (-1.0 / (F / ((Math.sin((Math.PI * l_m)) * (1.0 / (1.0 + (-0.5 * Math.pow((Math.PI * l_m), 2.0))))) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (-1.0 / (F / ((math.sin((math.pi * l_m)) * (1.0 / (1.0 + (-0.5 * math.pow((math.pi * l_m), 2.0))))) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(sin(Float64(pi * l_m)) * Float64(1.0 / Float64(1.0 + Float64(-0.5 * (Float64(pi * l_m) ^ 2.0))))) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (-1.0 / (F / ((sin((pi * l_m)) * (1.0 / (1.0 + (-0.5 * ((pi * l_m) ^ 2.0))))) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 + N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

   1. tan-quot 80.2%

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

   2. div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in l around 0 92.3%

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

   1. *-commutative 92.3%

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

   2. unpow2 92.3%

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

   3. unpow2 92.3%

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

   4. swap-sqr 92.3%

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

   5. unpow2 92.3%

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

   6. *-commutative 92.3%

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

9. Simplified 92.3%

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

10. Final simplification 92.3%

    \[\leadsto \pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + -0.5 \cdot {\left(\pi \cdot \ell\right)}^{2}}}{F}}} \]
11. Add Preprocessing

Alternative 7: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\pi \cdot l\_m}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-24)
    (+ (* PI l_m) (/ -1.0 (/ F (/ (* PI l_m) F))))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e-24) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F / ((((double) M_PI) * l_m) / F)));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 2e-24) {
		tmp = (Math.PI * l_m) + (-1.0 / (F / ((Math.PI * l_m) / F)));
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 2e-24:
		tmp = (math.pi * l_m) + (-1.0 / (F / ((math.pi * l_m) / F)))
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e-24)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(Float64(pi * l_m) / F))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e-24)
		tmp = (pi * l_m) + (-1.0 / (F / ((pi * l_m) / F)));
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-24], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.99999999999999985e-24

   1. Initial program 81.5%

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

      1. associate-*l/ 81.6%

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

      2. *-un-lft-identity 81.6%

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

      3. associate-/r* 87.6%

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

      4. clear-num 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in l around 0 81.6%

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

   if 1.99999999999999985e-24 < (*.f64 (PI.f64) l)

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. sqr-neg 61.9%

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

      3. associate-*r/ 61.9%

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

      4. sqr-neg 61.9%

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

      5. *-rgt-identity 61.9%

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

   3. Simplified 61.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\pi \cdot \ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 1.0× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (/ -1.0 (/ F (/ (tan (* PI l_m)) F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + (-1.0 / (F / (tan((((double) M_PI) * l_m)) / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + (-1.0 / (F / (Math.tan((Math.PI * l_m)) / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + (-1.0 / (F / (math.tan((math.pi * l_m)) / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(tan(Float64(pi * l_m)) / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + (-1.0 / (F / (tan((pi * l_m)) / F))));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

   4. clear-num 80.2%

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

4. Applied egg-rr 80.2%

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

5. Final simplification 80.2%

   \[\leadsto \pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}} \]
6. Add Preprocessing

Alternative 9: 82.4% accurate, 1.0× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((tan((pi * l_m)) / F) / F));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-*l/ 75.9%

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

   2. *-un-lft-identity 75.9%

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

   3. associate-/r* 80.1%

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

4. Applied egg-rr 80.1%

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

Alternative 10: 74.7% accurate, 8.7× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (* (* PI (/ l_m F)) (/ -1.0 F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((((double) M_PI) * (l_m / F)) * (-1.0 / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + ((Math.PI * (l_m / F)) * (-1.0 / F)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + ((math.pi * (l_m / F)) * (-1.0 / F)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(pi * Float64(l_m / F)) * Float64(-1.0 / F))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + ((pi * (l_m / F)) * (-1.0 / F)));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. sqr-neg 75.8%

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

   3. associate-*r/ 75.9%

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

   4. sqr-neg 75.9%

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

   5. *-rgt-identity 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. *-un-lft-identity 69.0%

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

   3. associate-*r/ 69.0%

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

   4. times-frac 73.3%

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

   5. associate-/l* 73.3%

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

7. Applied egg-rr 73.3%

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

8. Final simplification 73.3%

   \[\leadsto \pi \cdot \ell + \left(\pi \cdot \frac{\ell}{F}\right) \cdot \frac{-1}{F} \]
9. Add Preprocessing

Alternative 11: 74.7% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ PI F) (/ F l_m)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((((double) M_PI) / F) / (F / l_m)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.PI / F) / (F / l_m)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.pi / F) / (F / l_m)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(pi / F) / Float64(F / l_m))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((pi / F) / (F / l_m)));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. sqr-neg 75.8%

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

   3. associate-*r/ 75.9%

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

   4. sqr-neg 75.9%

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

   5. *-rgt-identity 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. times-frac 73.3%

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

7. Applied egg-rr 73.3%

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

   1. clear-num 73.3%

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

   2. un-div-inv 73.3%

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

9. Applied egg-rr 73.3%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\pi}{F}}{\frac{F}{\ell}}} \]
10. Add Preprocessing

Alternative 12: 74.7% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ PI (/ F (/ l_m F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - (((double) M_PI) / (F / (l_m / F))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - (Math.PI / (F / (l_m / F))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - (math.pi / (F / (l_m / F))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(pi / Float64(F / Float64(l_m / F)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - (pi / (F / (l_m / F))));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. sqr-neg 75.8%

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

   3. associate-*r/ 75.9%

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

   4. sqr-neg 75.9%

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

   5. *-rgt-identity 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. times-frac 73.3%

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

7. Applied egg-rr 73.3%

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

   1. *-commutative 73.3%

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

   2. clear-num 73.3%

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

   3. frac-times 73.3%

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

   4. *-un-lft-identity 73.3%

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

9. Applied egg-rr 73.3%

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

    1. *-commutative 73.3%

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

    2. clear-num 73.3%

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

    3. un-div-inv 73.3%

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

11. Applied egg-rr 73.3%

    \[\leadsto \pi \cdot \ell - \frac{\pi}{\color{blue}{\frac{F}{\frac{\ell}{F}}}} \]
12. Add Preprocessing

Alternative 13: 74.7% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ l_m F) (/ PI F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((l_m / F) * (Math.PI / F)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((l_m / F) * (math.pi / F)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((l_m / F) * (pi / F)));
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. sqr-neg 75.8%

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

   3. associate-*r/ 75.9%

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

   4. sqr-neg 75.9%

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

   5. *-rgt-identity 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. times-frac 73.3%

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

7. Applied egg-rr 73.3%

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

8. Final simplification 73.3%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))