Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 90.8%

Time: 18.9s

Alternatives: 15

Speedup: 3.8×

• Could not converge on a ground truth

  1. t = 5.33573598492934e-294
  2. l = 6.810743392652214
  3. k = 1.3897592430329282e+246

• 39.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k}\\ t_3 := \sin k \cdot \tan k\\ t_4 := \sqrt[3]{t\_3}\\ t_5 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_3 \cdot {\left(k \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{t\_2 \cdot \left(t\_m \cdot {\left(t\_4 \cdot \left(t\_m \cdot t\_5\right)\right)}^{-2}\right)}\right)}^{3} \cdot \left(t\_2 \cdot \frac{\frac{1}{t\_5}}{t\_4}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k))
        (t_3 (* (sin k) (tan k)))
        (t_4 (cbrt t_3))
        (t_5 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= (* l l) 1e-317)
      (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+307)
        (* (* l l) (/ 2.0 (* t_3 (pow (* k (sqrt t_m)) 2.0))))
        (*
         (pow (cbrt (* t_2 (* t_m (pow (* t_4 (* t_m t_5)) -2.0)))) 3.0)
         (* t_2 (/ (/ 1.0 t_5) t_4))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sqrt(2.0) / k;
	double t_3 = sin(k) * tan(k);
	double t_4 = cbrt(t_3);
	double t_5 = pow(cbrt(l), -2.0);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_3 * pow((k * sqrt(t_m)), 2.0)));
	} else {
		tmp = pow(cbrt((t_2 * (t_m * pow((t_4 * (t_m * t_5)), -2.0)))), 3.0) * (t_2 * ((1.0 / t_5) / t_4));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = Math.sin(k) * Math.tan(k);
	double t_4 = Math.cbrt(t_3);
	double t_5 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_3 * Math.pow((k * Math.sqrt(t_m)), 2.0)));
	} else {
		tmp = Math.pow(Math.cbrt((t_2 * (t_m * Math.pow((t_4 * (t_m * t_5)), -2.0)))), 3.0) * (t_2 * ((1.0 / t_5) / t_4));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = Float64(sin(k) * tan(k))
	t_4 = cbrt(t_3)
	t_5 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e-317)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+307)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_3 * (Float64(k * sqrt(t_m)) ^ 2.0))));
	else
		tmp = Float64((cbrt(Float64(t_2 * Float64(t_m * (Float64(t_4 * Float64(t_m * t_5)) ^ -2.0)))) ^ 3.0) * Float64(t_2 * Float64(Float64(1.0 / t_5) / t_4)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 1/3], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-317], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+307], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$3 * N[Power[N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(t$95$2 * N[(t$95$m * N[Power[N[(t$95$4 * N[(t$95$m * t$95$5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(t$95$2 * N[(N[(1.0 / t$95$5), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000023e-317

   1. Initial program 31.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 42.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 29.8%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 29.8%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 43.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 1.00000023e-317 < (*.f64 l l) < 9.99999999999999986e306

   1. Initial program 44.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 28.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 28.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 26.0%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 26.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 28.1%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 32.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 32.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 33.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 33.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 33.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 37.8%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 37.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 37.9%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 47.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 47.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   if 9.99999999999999986e306 < (*.f64 l l)

   1. Initial program 20.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 83.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-/l* 83.8%

          \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 91.2%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right)\right)} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. associate-*l* 91.1%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. associate-/l* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       5. associate-/r* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\color{blue}{\frac{\frac{t}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       6. *-inverses 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{\color{blue}{1}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 91.2%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 83.8%

          \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       2. associate-*r* 83.8%

          \[\leadsto \left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\color{blue}{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       3. add-cube-cbrt 83.9%

          \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}} \cdot \sqrt[3]{\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}}\right) \cdot \sqrt[3]{\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}}\right)} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. pow3 83.9%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   14. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right)}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right)}\right)}^{3} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_4 := \sin k \cdot \tan k\\ t_5 := \sqrt[3]{t\_4}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_4 \cdot {\left(k \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\frac{1}{t\_3}}{t\_5}\right) \cdot \left(t\_2 \cdot \left(t\_m \cdot {\left(t\_m \cdot \left(t\_5 \cdot t\_3\right)\right)}^{-2}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k))
        (t_3 (pow (cbrt l) -2.0))
        (t_4 (* (sin k) (tan k)))
        (t_5 (cbrt t_4)))
   (*
    t_s
    (if (<= (* l l) 1e-317)
      (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+307)
        (* (* l l) (/ 2.0 (* t_4 (pow (* k (sqrt t_m)) 2.0))))
        (*
         (* t_2 (/ (/ 1.0 t_3) t_5))
         (* t_2 (* t_m (pow (* t_m (* t_5 t_3)) -2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sqrt(2.0) / k;
	double t_3 = pow(cbrt(l), -2.0);
	double t_4 = sin(k) * tan(k);
	double t_5 = cbrt(t_4);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_4 * pow((k * sqrt(t_m)), 2.0)));
	} else {
		tmp = (t_2 * ((1.0 / t_3) / t_5)) * (t_2 * (t_m * pow((t_m * (t_5 * t_3)), -2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = Math.pow(Math.cbrt(l), -2.0);
	double t_4 = Math.sin(k) * Math.tan(k);
	double t_5 = Math.cbrt(t_4);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_4 * Math.pow((k * Math.sqrt(t_m)), 2.0)));
	} else {
		tmp = (t_2 * ((1.0 / t_3) / t_5)) * (t_2 * (t_m * Math.pow((t_m * (t_5 * t_3)), -2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = cbrt(l) ^ -2.0
	t_4 = Float64(sin(k) * tan(k))
	t_5 = cbrt(t_4)
	tmp = 0.0
	if (Float64(l * l) <= 1e-317)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+307)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_4 * (Float64(k * sqrt(t_m)) ^ 2.0))));
	else
		tmp = Float64(Float64(t_2 * Float64(Float64(1.0 / t_3) / t_5)) * Float64(t_2 * Float64(t_m * (Float64(t_m * Float64(t_5 * t_3)) ^ -2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$4, 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-317], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+307], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$4 * N[Power[N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t$95$m * N[Power[N[(t$95$m * N[(t$95$5 * t$95$3), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000023e-317

   1. Initial program 31.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 42.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 29.8%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 29.8%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 43.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 1.00000023e-317 < (*.f64 l l) < 9.99999999999999986e306

   1. Initial program 44.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 28.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 28.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 26.0%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 26.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 28.1%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 32.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 32.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 33.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 33.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 33.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 37.8%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 37.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 37.9%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 47.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 47.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   if 9.99999999999999986e306 < (*.f64 l l)

   1. Initial program 20.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 83.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-/l* 83.8%

          \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 91.2%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right)\right)} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. associate-*l* 91.1%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. associate-/l* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       5. associate-/r* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\color{blue}{\frac{\frac{t}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       6. *-inverses 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{\color{blue}{1}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 91.2%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \tan k\\ t_3 := \sqrt[3]{t\_2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_2 \cdot {\left(k \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \left(t\_m \cdot {\left(t\_m \cdot \left(t\_3 \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right)\right) \cdot \frac{\sqrt{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{k \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) (tan k))) (t_3 (cbrt t_2)))
   (*
    t_s
    (if (<= (* l l) 1e-317)
      (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+307)
        (* (* l l) (/ 2.0 (* t_2 (pow (* k (sqrt t_m)) 2.0))))
        (*
         (*
          (/ (sqrt 2.0) k)
          (* t_m (pow (* t_m (* t_3 (pow (cbrt l) -2.0))) -2.0)))
         (/ (* (sqrt 2.0) (pow (cbrt l) 2.0)) (* k t_3))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * tan(k);
	double t_3 = cbrt(t_2);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_2 * pow((k * sqrt(t_m)), 2.0)));
	} else {
		tmp = ((sqrt(2.0) / k) * (t_m * pow((t_m * (t_3 * pow(cbrt(l), -2.0))), -2.0))) * ((sqrt(2.0) * pow(cbrt(l), 2.0)) / (k * t_3));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * Math.tan(k);
	double t_3 = Math.cbrt(t_2);
	double tmp;
	if ((l * l) <= 1e-317) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+307) {
		tmp = (l * l) * (2.0 / (t_2 * Math.pow((k * Math.sqrt(t_m)), 2.0)));
	} else {
		tmp = ((Math.sqrt(2.0) / k) * (t_m * Math.pow((t_m * (t_3 * Math.pow(Math.cbrt(l), -2.0))), -2.0))) * ((Math.sqrt(2.0) * Math.pow(Math.cbrt(l), 2.0)) / (k * t_3));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * tan(k))
	t_3 = cbrt(t_2)
	tmp = 0.0
	if (Float64(l * l) <= 1e-317)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+307)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_2 * (Float64(k * sqrt(t_m)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / k) * Float64(t_m * (Float64(t_m * Float64(t_3 * (cbrt(l) ^ -2.0))) ^ -2.0))) * Float64(Float64(sqrt(2.0) * (cbrt(l) ^ 2.0)) / Float64(k * t_3)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-317], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+307], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$2 * N[Power[N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t$95$m * N[Power[N[(t$95$m * N[(t$95$3 * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000023e-317

   1. Initial program 31.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 42.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 29.8%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 29.8%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 43.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 1.00000023e-317 < (*.f64 l l) < 9.99999999999999986e306

   1. Initial program 44.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 28.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 28.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 26.0%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 26.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 28.1%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 30.5%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 32.0%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 32.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 32.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 33.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 33.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 33.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 33.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 37.8%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 37.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 37.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 37.9%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 47.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 47.8%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 47.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   if 9.99999999999999986e306 < (*.f64 l l)

   1. Initial program 20.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 83.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-/l* 83.8%

          \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 91.2%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right)\right)} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. associate-*l* 91.1%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. associate-/l* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       5. associate-/r* 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\color{blue}{\frac{\frac{t}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       6. *-inverses 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{\color{blue}{1}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 91.2%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
   13. Step-by-step derivation

       1. frac-times 91.2%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \color{blue}{\frac{\sqrt{2} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]

       2. pow-flip 91.1%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\sqrt{2} \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(--2\right)}}}{k \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

       3. metadata-eval 91.1%

          \[\leadsto \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\sqrt{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{2}}}{k \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   14. Applied egg-rr 91.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right)\right) \cdot \frac{\sqrt{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{k \cdot \sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \tan k\\ t_3 := \sqrt[3]{t\_2}\\ t_4 := \frac{\sqrt{2}}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-146}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{+153}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_2 \cdot {\left(k \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(t\_3 \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2} \cdot \left(t\_m \cdot t\_4\right)\right) \cdot \left(t\_4 \cdot {\ell}^{0.6666666666666666}\right)}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) (tan k))) (t_3 (cbrt t_2)) (t_4 (/ (sqrt 2.0) k)))
   (*
    t_s
    (if (<= l 4.5e-146)
      (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= l 5.9e+153)
        (* (* l l) (/ 2.0 (* t_2 (pow (* k (sqrt t_m)) 2.0))))
        (/
         (*
          (* (pow (* t_3 (* t_m (pow (cbrt l) -2.0))) -2.0) (* t_m t_4))
          (* t_4 (pow l 0.6666666666666666)))
         t_3))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * tan(k);
	double t_3 = cbrt(t_2);
	double t_4 = sqrt(2.0) / k;
	double tmp;
	if (l <= 4.5e-146) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 5.9e+153) {
		tmp = (l * l) * (2.0 / (t_2 * pow((k * sqrt(t_m)), 2.0)));
	} else {
		tmp = ((pow((t_3 * (t_m * pow(cbrt(l), -2.0))), -2.0) * (t_m * t_4)) * (t_4 * pow(l, 0.6666666666666666))) / t_3;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * Math.tan(k);
	double t_3 = Math.cbrt(t_2);
	double t_4 = Math.sqrt(2.0) / k;
	double tmp;
	if (l <= 4.5e-146) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 5.9e+153) {
		tmp = (l * l) * (2.0 / (t_2 * Math.pow((k * Math.sqrt(t_m)), 2.0)));
	} else {
		tmp = ((Math.pow((t_3 * (t_m * Math.pow(Math.cbrt(l), -2.0))), -2.0) * (t_m * t_4)) * (t_4 * Math.pow(l, 0.6666666666666666))) / t_3;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * tan(k))
	t_3 = cbrt(t_2)
	t_4 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (l <= 4.5e-146)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l <= 5.9e+153)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_2 * (Float64(k * sqrt(t_m)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64((Float64(t_3 * Float64(t_m * (cbrt(l) ^ -2.0))) ^ -2.0) * Float64(t_m * t_4)) * Float64(t_4 * (l ^ 0.6666666666666666))) / t_3);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 4.5e-146], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 5.9e+153], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$2 * N[Power[N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(t$95$3 * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(t$95$m * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[Power[l, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.5000000000000001e-146

   1. Initial program 34.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 29.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 29.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 25.6%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 36.8%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 4.5000000000000001e-146 < l < 5.9000000000000002e153

   1. Initial program 41.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 35.0%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 35.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 35.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 31.9%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 32.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 34.7%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 34.9%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 34.9%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 34.9%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 36.4%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 36.4%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 36.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 38.0%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 39.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 39.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 39.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 39.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 46.8%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 46.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 46.8%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 46.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 46.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 54.4%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 54.4%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 54.4%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 54.4%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 54.4%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.9000000000000002e153 < l

   1. Initial program 23.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 77.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 77.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 77.3%

       \[\leadsto \color{blue}{\frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-/r* 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \color{blue}{\frac{\frac{t}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       2. pow1 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{\color{blue}{{t}^{1}}}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. pow1 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\frac{{t}^{1}}{\color{blue}{{t}^{1}}}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. pow-div 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\color{blue}{{t}^{\left(1 - 1\right)}}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       5. metadata-eval 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{t}^{\color{blue}{0}}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       6. metadata-eval 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\color{blue}{1}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       7. pow-flip 77.3%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(--2\right)}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       8. pow1/3 76.5%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{\left(--2\right)}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       9. metadata-eval 76.5%

          \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left({\ell}^{0.3333333333333333}\right)}^{\color{blue}{2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       10. pow-pow 76.5%

           \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \color{blue}{{\ell}^{\left(0.3333333333333333 \cdot 2\right)}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

       11. metadata-eval 76.5%

           \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot {\ell}^{\color{blue}{0.6666666666666666}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

   12. Applied egg-rr 76.5%

       \[\leadsto \frac{\left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \color{blue}{{\ell}^{0.6666666666666666}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-146}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{+153}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2} \cdot \left(t \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot {\ell}^{0.6666666666666666}\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-18)
    (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (* (* l l) (/ 2.0 (* (* (sin k) (tan k)) (pow (* k (sqrt t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((sin(k) * tan(k)) * pow((k * sqrt(t_m)), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-18) then
        tmp = (((l * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((sin(k) * tan(k)) * ((k * sqrt(t_m)) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow((k * Math.sqrt(t_m)), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-18:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l * l) * (2.0 / ((math.sin(k) * math.tan(k)) * math.pow((k * math.sqrt(t_m)), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-18)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(k * sqrt(t_m)) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-18)
		tmp = (((l * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((sin(k) * tan(k)) * ((k * sqrt(t_m)) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-18], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999987e-18

   1. Initial program 37.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.1%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 41.6%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 8.99999999999999987e-18 < k

   1. Initial program 30.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 21.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 21.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 12.6%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 15.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 15.2%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 16.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 16.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 21.2%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 21.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 21.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 36.1%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 36.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-18)
    (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (/ (* 2.0 (pow l 2.0)) (* (* (sin k) (tan k)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (2.0 * pow(l, 2.0)) / ((sin(k) * tan(k)) * (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-18) then
        tmp = (((l * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / ((sin(k) * tan(k)) * ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / ((Math.sin(k) * Math.tan(k)) * (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-18:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / ((math.sin(k) * math.tan(k)) * (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-18)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(Float64(sin(k) * tan(k)) * Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-18)
		tmp = (((l * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (2.0 * (l ^ 2.0)) / ((sin(k) * tan(k)) * ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-18], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999987e-18

   1. Initial program 37.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.1%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 41.6%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 8.99999999999999987e-18 < k

   1. Initial program 30.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 21.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 21.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 12.6%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 15.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 15.2%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 16.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 16.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \]

      2. pow2 16.5%

         \[\leadsto \frac{2 \cdot \color{blue}{{\ell}^{2}}}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}} \]

      3. unpow-prod-down 16.5%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \]

      4. pow2 16.5%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k \cdot {t}^{1.5}}{t}\right)}^{2}} \]

      5. add-sqr-sqrt 30.2%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k \cdot {t}^{1.5}}{t}\right)}^{2}} \]

      6. associate-/l* 30.2%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}}^{2}} \]

      7. unpow-prod-down 30.0%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({k}^{2} \cdot {\left(\frac{{t}^{1.5}}{t}\right)}^{2}\right)}} \]

      8. pow1 30.0%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot {\left(\frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)}^{2}\right)} \]

      9. pow-div 34.9%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot {\color{blue}{\left({t}^{\left(1.5 - 1\right)}\right)}}^{2}\right)} \]

      10. metadata-eval 34.9%

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot {\left({t}^{\color{blue}{0.5}}\right)}^{2}\right)} \]

      11. pow1/2 34.9%

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot {\color{blue}{\left(\sqrt{t}\right)}}^{2}\right)} \]

      12. pow2 34.9%

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right)} \]

      13. add-sqr-sqrt 73.3%

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   10. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-18)
    (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (* (* l l) (/ (/ (/ 2.0 (sin k)) (tan k)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-18) then
        tmp = (((l * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l * l) * (((2.0d0 / sin(k)) / tan(k)) / ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / Math.sin(k)) / Math.tan(k)) / (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-18:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l * l) * (((2.0 / math.sin(k)) / math.tan(k)) / (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-18)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(2.0 / sin(k)) / tan(k)) / Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-18)
		tmp = (((l * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-18], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999987e-18

   1. Initial program 37.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.1%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 41.6%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 8.99999999999999987e-18 < k

   1. Initial program 30.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 21.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 21.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 12.6%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 15.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 15.2%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 16.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 16.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 21.2%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 21.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 21.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 36.1%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 36.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   13. Step-by-step derivation

       1. div-inv 36.1%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

       2. *-commutative 36.1%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left(\sqrt{t} \cdot k\right)}}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       3. unpow-prod-down 34.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left(\sqrt{t}\right)}^{2} \cdot {k}^{2}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

       4. pow2 34.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. add-sqr-sqrt 73.2%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{t} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. Applied egg-rr 73.2%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   15. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 73.2%

          \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. associate-/r* 73.3%

          \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

       4. associate-/r* 73.3%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}{t \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   16. Simplified 73.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{{k}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-18)
    (pow (* l (* (sqrt 2.0) (/ (sqrt (/ 1.0 t_m)) (pow k 2.0)))) 2.0)
    (* (* l l) (/ (/ (/ 2.0 (sin k)) (tan k)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = pow((l * (sqrt(2.0) * (sqrt((1.0 / t_m)) / pow(k, 2.0)))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-18) then
        tmp = (l * (sqrt(2.0d0) * (sqrt((1.0d0 / t_m)) / (k ** 2.0d0)))) ** 2.0d0
    else
        tmp = (l * l) * (((2.0d0 / sin(k)) / tan(k)) / ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = Math.pow((l * (Math.sqrt(2.0) * (Math.sqrt((1.0 / t_m)) / Math.pow(k, 2.0)))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / Math.sin(k)) / Math.tan(k)) / (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-18:
		tmp = math.pow((l * (math.sqrt(2.0) * (math.sqrt((1.0 / t_m)) / math.pow(k, 2.0)))), 2.0)
	else:
		tmp = (l * l) * (((2.0 / math.sin(k)) / math.tan(k)) / (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-18)
		tmp = Float64(l * Float64(sqrt(2.0) * Float64(sqrt(Float64(1.0 / t_m)) / (k ^ 2.0)))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(2.0 / sin(k)) / tan(k)) / Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-18)
		tmp = (l * (sqrt(2.0) * (sqrt((1.0 / t_m)) / (k ^ 2.0)))) ^ 2.0;
	else
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-18], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999987e-18

   1. Initial program 37.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.1%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 41.6%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-*l/ 41.1%

          \[\leadsto {\color{blue}{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]

       2. associate-*l* 41.1%

          \[\leadsto {\left(\frac{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{t}}\right)}}{{k}^{2}}\right)}^{2} \]

       3. *-commutative 41.1%

          \[\leadsto {\left(\frac{\ell \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot \sqrt{2}\right)}}{{k}^{2}}\right)}^{2} \]

       4. associate-/l* 41.2%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{\sqrt{\frac{1}{t}} \cdot \sqrt{2}}{{k}^{2}}\right)}}^{2} \]

       5. *-commutative 41.2%

          \[\leadsto {\left(\ell \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{t}}}}{{k}^{2}}\right)}^{2} \]

       6. associate-/l* 41.2%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left(\sqrt{2} \cdot \frac{\sqrt{\frac{1}{t}}}{{k}^{2}}\right)}\right)}^{2} \]

   11. Simplified 41.2%

       \[\leadsto {\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\sqrt{\frac{1}{t}}}{{k}^{2}}\right)\right)}}^{2} \]

   if 8.99999999999999987e-18 < k

   1. Initial program 30.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 21.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 21.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 12.6%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 12.6%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 14.0%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 15.1%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 15.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 15.2%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 16.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 16.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 16.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 21.2%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 21.2%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 21.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 21.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 36.1%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 36.1%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 36.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   13. Step-by-step derivation

       1. div-inv 36.1%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

       2. *-commutative 36.1%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left(\sqrt{t} \cdot k\right)}}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       3. unpow-prod-down 34.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left(\sqrt{t}\right)}^{2} \cdot {k}^{2}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

       4. pow2 34.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. add-sqr-sqrt 73.2%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{t} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. Applied egg-rr 73.2%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   15. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 73.2%

          \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. associate-/r* 73.3%

          \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

       4. associate-/r* 73.3%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}{t \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   16. Simplified 73.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\sqrt{\frac{1}{t}}}{{k}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k}}{\frac{k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 5.4e-177)
    (pow (* l (/ (/ (sqrt (/ 2.0 (pow t_m 3.0))) k) (/ k t_m))) 2.0)
    (* (* l l) (/ (/ (/ 2.0 (sin k)) (tan k)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5.4e-177) {
		tmp = pow((l * ((sqrt((2.0 / pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 5.4d-177) then
        tmp = (l * ((sqrt((2.0d0 / (t_m ** 3.0d0))) / k) / (k / t_m))) ** 2.0d0
    else
        tmp = (l * l) * (((2.0d0 / sin(k)) / tan(k)) / ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5.4e-177) {
		tmp = Math.pow((l * ((Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / Math.sin(k)) / Math.tan(k)) / (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 5.4e-177:
		tmp = math.pow((l * ((math.sqrt((2.0 / math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0)
	else:
		tmp = (l * l) * (((2.0 / math.sin(k)) / math.tan(k)) / (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 5.4e-177)
		tmp = Float64(l * Float64(Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k) / Float64(k / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(2.0 / sin(k)) / tan(k)) / Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 5.4e-177)
		tmp = (l * ((sqrt((2.0 / (t_m ^ 3.0))) / k) / (k / t_m))) ^ 2.0;
	else
		tmp = (l * l) * (((2.0 / sin(k)) / tan(k)) / ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 5.4e-177], N[Power[N[(l * N[(N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.4000000000000004e-177

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.4%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 25.6%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 34.6%

      \[\leadsto {\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\color{blue}{k}}}{\frac{k}{t}}\right)}^{2} \]

   if 5.4000000000000004e-177 < l

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 28.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative 28.2%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {t}^{3}}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      4. sqrt-prod 25.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-*r* 25.1%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      6. sqrt-prod 27.0%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      7. sqrt-pow1 29.2%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      8. metadata-eval 29.2%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot {\left(\frac{k}{t}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      9. pow1 29.2%

         \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \sqrt{{t}^{3}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      10. sqrt-pow1 31.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 31.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*l* 32.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)\right)}}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-*l/ 33.4%

         \[\leadsto \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\frac{k \cdot {t}^{1.5}}{t}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   8. Simplified 33.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. unpow2 33.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/l* 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. pow1 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-div 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      6. pow1/2 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k \cdot {t}^{1.5}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-/l* 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \color{blue}{\left(k \cdot \frac{{t}^{1.5}}{t}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      8. pow1 33.4%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      9. pow-div 40.5%

         \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      10. metadata-eval 40.5%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot {t}^{\color{blue}{0.5}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      11. pow1/2 40.5%

          \[\leadsto \frac{2}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \color{blue}{\sqrt{t}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 40.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. swap-sqr 40.5%

          \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. rem-square-sqrt 48.9%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(k \cdot \sqrt{t}\right) \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. unpow1 48.9%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{1}} \cdot \left(k \cdot \sqrt{t}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

       4. pow-plus 48.9%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{{\left(k \cdot \sqrt{t}\right)}^{\left(1 + 1\right)}}} \cdot \left(\ell \cdot \ell\right) \]

       5. metadata-eval 48.9%

          \[\leadsto \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{\color{blue}{2}}} \cdot \left(\ell \cdot \ell\right) \]

   12. Simplified 48.9%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   13. Step-by-step derivation

       1. div-inv 48.9%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\left(k \cdot \sqrt{t}\right)}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

       2. *-commutative 48.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left(\sqrt{t} \cdot k\right)}}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       3. unpow-prod-down 45.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left(\sqrt{t}\right)}^{2} \cdot {k}^{2}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

       4. pow2 45.9%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. add-sqr-sqrt 84.6%

          \[\leadsto \left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{t} \cdot {k}^{2}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. Applied egg-rr 84.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   15. Step-by-step derivation

       1. associate-*r/ 84.6%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 84.6%

          \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right) \]

       3. associate-/r* 84.6%

          \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

       4. associate-/r* 84.6%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}{t \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   16. Simplified 84.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{k}}{\frac{k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k}}{\frac{k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.65e-178)
    (pow (* l (/ (/ (sqrt (/ 2.0 (pow t_m 3.0))) k) (/ k t_m))) 2.0)
    (* (* l l) (/ (/ 2.0 (* (sin k) (tan k))) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.65e-178) {
		tmp = pow((l * ((sqrt((2.0 / pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / (sin(k) * tan(k))) / (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.65d-178) then
        tmp = (l * ((sqrt((2.0d0 / (t_m ** 3.0d0))) / k) / (k / t_m))) ** 2.0d0
    else
        tmp = (l * l) * ((2.0d0 / (sin(k) * tan(k))) / ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.65e-178) {
		tmp = Math.pow((l * ((Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / (Math.sin(k) * Math.tan(k))) / (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.65e-178:
		tmp = math.pow((l * ((math.sqrt((2.0 / math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 / (math.sin(k) * math.tan(k))) / (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.65e-178)
		tmp = Float64(l * Float64(Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k) / Float64(k / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.65e-178)
		tmp = (l * ((sqrt((2.0 / (t_m ^ 3.0))) / k) / (k / t_m))) ^ 2.0;
	else
		tmp = (l * l) * ((2.0 / (sin(k) * tan(k))) / ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.65e-178], N[Power[N[(l * N[(N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.6500000000000001e-178

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.4%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 25.6%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 34.6%

      \[\leadsto {\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\color{blue}{k}}}{\frac{k}{t}}\right)}^{2} \]

   if 1.6500000000000001e-178 < l

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 29.8%

         \[\leadsto \frac{2}{\color{blue}{\log \left(e^{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. exp-prod 37.5%

         \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 37.5%

      \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around inf 84.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. associate-*r* 84.6%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. *-commutative 84.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}} \cdot \left(\ell \cdot \ell\right) \]

      3. associate-/r* 84.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 84.6%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \tan k}}{\color{blue}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   9. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{k}}{\frac{k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k}}{\frac{k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t\_m \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.5e-178)
    (pow (* l (/ (/ (sqrt (/ 2.0 (pow t_m 3.0))) k) (/ k t_m))) 2.0)
    (* (* l l) (/ 2.0 (* (pow k 2.0) (* t_m (* (sin k) (tan k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.5e-178) {
		tmp = pow((l * ((sqrt((2.0 / pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k, 2.0) * (t_m * (sin(k) * tan(k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.5d-178) then
        tmp = (l * ((sqrt((2.0d0 / (t_m ** 3.0d0))) / k) / (k / t_m))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((k ** 2.0d0) * (t_m * (sin(k) * tan(k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.5e-178) {
		tmp = Math.pow((l * ((Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k, 2.0) * (t_m * (Math.sin(k) * Math.tan(k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.5e-178:
		tmp = math.pow((l * ((math.sqrt((2.0 / math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k, 2.0) * (t_m * (math.sin(k) * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.5e-178)
		tmp = Float64(l * Float64(Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k) / Float64(k / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * Float64(sin(k) * tan(k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.5e-178)
		tmp = (l * ((sqrt((2.0 / (t_m ^ 3.0))) / k) / (k / t_m))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((k ^ 2.0) * (t_m * (sin(k) * tan(k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.5e-178], N[Power[N[(l * N[(N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.4999999999999999e-178

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.4%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 25.6%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 34.6%

      \[\leadsto {\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\color{blue}{k}}}{\frac{k}{t}}\right)}^{2} \]

   if 1.4999999999999999e-178 < l

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 29.8%

         \[\leadsto \frac{2}{\color{blue}{\log \left(e^{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. exp-prod 37.5%

         \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 37.5%

      \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around inf 84.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{k}}{\frac{k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k}}{\frac{k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{{k}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.45e-178)
    (pow (* l (/ (/ (sqrt (/ 2.0 (pow t_m 3.0))) k) (/ k t_m))) 2.0)
    (* (* l l) (/ (/ 2.0 (pow k 2.0)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.45e-178) {
		tmp = pow((l * ((sqrt((2.0 / pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) / (pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.45d-178) then
        tmp = (l * ((sqrt((2.0d0 / (t_m ** 3.0d0))) / k) / (k / t_m))) ** 2.0d0
    else
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) / ((k ** 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.45e-178) {
		tmp = Math.pow((l * ((Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) / (Math.pow(k, 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.45e-178:
		tmp = math.pow((l * ((math.sqrt((2.0 / math.pow(t_m, 3.0))) / k) / (k / t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) / (math.pow(k, 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.45e-178)
		tmp = Float64(l * Float64(Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k) / Float64(k / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) / Float64((k ^ 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.45e-178)
		tmp = (l * ((sqrt((2.0 / (t_m ^ 3.0))) / k) / (k / t_m))) ^ 2.0;
	else
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) / ((k ^ 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.45e-178], N[Power[N[(l * N[(N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.4499999999999999e-178

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.4%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow2 28.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)}\right)}^{2}} \]

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-/r* 25.6%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}}\right)}^{2} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 34.6%

      \[\leadsto {\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{\color{blue}{k}}}{\frac{k}{t}}\right)}^{2} \]

   if 1.4499999999999999e-178 < l

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 29.8%

         \[\leadsto \frac{2}{\color{blue}{\log \left(e^{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. exp-prod 37.5%

         \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 37.5%

      \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around inf 84.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. associate-*r* 84.6%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \left(\ell \cdot \ell\right) \]

      2. *-commutative 84.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}} \cdot \left(\ell \cdot \ell\right) \]

      3. associate-/r* 84.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 84.6%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \tan k}}{\color{blue}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   9. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   10. Taylor expanded in k around 0 65.5%

       \[\leadsto \frac{\color{blue}{\frac{2}{{k}^{2}}}}{t \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{\frac{2}{{t}^{3}}}}{k}}{\frac{k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{{k}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{{k}^{2} \cdot t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ (/ 2.0 (pow k 2.0)) (* (pow k 2.0) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / pow(k, 2.0)) / (pow(k, 2.0) * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((2.0d0 / (k ** 2.0d0)) / ((k ** 2.0d0) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / Math.pow(k, 2.0)) / (Math.pow(k, 2.0) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * ((2.0 / math.pow(k, 2.0)) / (math.pow(k, 2.0) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) / Float64((k ^ 2.0) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * ((2.0 / (k ^ 2.0)) / ((k ^ 2.0) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.8%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 29.0%

      \[\leadsto \frac{2}{\color{blue}{\log \left(e^{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

   2. exp-prod 34.3%

      \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

6. Applied egg-rr 34.3%

   \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{{t}^{3}}\right)}^{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}} \cdot \left(\ell \cdot \ell\right) \]

7. Taylor expanded in k around inf 77.1%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. associate-*r* 77.1%

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \left(\ell \cdot \ell\right) \]

   2. *-commutative 77.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}} \cdot \left(\ell \cdot \ell\right) \]

   3. associate-/r* 77.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

   4. *-commutative 77.1%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \tan k}}{\color{blue}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

9. Simplified 77.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \tan k}}{t \cdot {k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

10. Taylor expanded in k around 0 65.9%

    \[\leadsto \frac{\color{blue}{\frac{2}{{k}^{2}}}}{t \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

11. Final simplification 65.9%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{{k}^{2} \cdot t} \]
12. Add Preprocessing

Alternative 14: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (t_m * pow(k, 4.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 / (t_m * (k ** 4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 / (t_m * math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 / (t_m * (k ^ 4.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.8%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 62.9%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 15: 62.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t\_m} \cdot {k}^{-4}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (* (/ 2.0 t_m) (pow k -4.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / t_m) * pow(k, -4.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((2.0d0 / t_m) * (k ** (-4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / t_m) * Math.pow(k, -4.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * ((2.0 / t_m) * math.pow(k, -4.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 / t_m) * (k ^ -4.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * ((2.0 / t_m) * (k ^ -4.0)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.8%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. *-commutative 62.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. associate-/r* 62.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 62.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. div-inv 62.8%

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 62.8%

      \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

   3. metadata-eval 62.8%

      \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]

10. Final simplification 62.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))