Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 94.0%

Time: 19.0s

Alternatives: 22

Speedup: 3.6×

• 40.8% 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: 35.5% 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: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 11200000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_3}}{\log \left(e^{t\_2}\right)}\right)}^{2}}{t\_m \cdot \left(t\_3 \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m)))) (t_3 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k_m 11200000000000.0)
      (pow
       (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/
       (pow (/ (/ (/ (sqrt 2.0) k_m) t_3) (log (exp t_2))) 2.0)
       (* t_m (* t_3 t_2)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = pow((((sqrt(2.0) / k_m) / t_3) / log(exp(t_2))), 2.0) / (t_m * (t_3 * t_2));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.pow((((Math.sqrt(2.0) / k_m) / t_3) / Math.log(Math.exp(t_2))), 2.0) / (t_m * (t_3 * t_2));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 11200000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_3) / log(exp(t_2))) ^ 2.0) / Float64(t_m * Float64(t_3 * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 11200000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] / N[Log[N[Exp[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e13

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.12e13 < k

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.6%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.2%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r/ 76.2%

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

   8. Simplified 76.2%

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

      1. frac-times 73.2%

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

      2. associate-/r/ 73.2%

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

      3. associate-/l* 73.3%

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

      4. div-inv 73.3%

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

      5. pow-flip 73.3%

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

      6. metadata-eval 73.3%

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

   10. Applied egg-rr 73.2%

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

       1. times-frac 76.2%

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

       2. associate-*r/ 76.2%

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

       3. associate-/r/ 76.2%

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

       4. associate-/r* 76.2%

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

       5. associate-*l/ 73.2%

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

   12. Simplified 90.0%

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

       1. add-log-exp 90.0%

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

   14. Applied egg-rr 90.0%

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

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 11200000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\sqrt{2}}{k\_m}}{t\_2 \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}}\right)}^{2}}{t\_m \cdot \left(t\_2 \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k_m 11200000000000.0)
      (pow
       (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/
       (pow (/ (/ (sqrt 2.0) k_m) (* t_2 (cbrt (* (sin k_m) (tan k_m))))) 2.0)
       (* t_m (* t_2 (* (cbrt (tan k_m)) (cbrt (sin k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = pow(((sqrt(2.0) / k_m) / (t_2 * cbrt((sin(k_m) * tan(k_m))))), 2.0) / (t_m * (t_2 * (cbrt(tan(k_m)) * cbrt(sin(k_m)))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / (t_2 * Math.cbrt((Math.sin(k_m) * Math.tan(k_m))))), 2.0) / (t_m * (t_2 * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m)))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 11200000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(sqrt(2.0) / k_m) / Float64(t_2 * cbrt(Float64(sin(k_m) * tan(k_m))))) ^ 2.0) / Float64(t_m * Float64(t_2 * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 11200000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$2 * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e13

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.12e13 < k

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.6%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.2%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r/ 76.2%

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

   8. Simplified 76.2%

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

      1. frac-times 73.2%

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

      2. associate-/r/ 73.2%

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

      3. associate-/l* 73.3%

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

      4. div-inv 73.3%

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

      5. pow-flip 73.3%

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

      6. metadata-eval 73.3%

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

   10. Applied egg-rr 73.2%

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

       1. times-frac 76.2%

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

       2. associate-*r/ 76.2%

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

       3. associate-/r/ 76.2%

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

       4. associate-/r* 76.2%

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

       5. associate-*l/ 73.2%

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

   12. Simplified 90.0%

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

       1. *-commutative 90.0%

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

       2. cbrt-prod 90.0%

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

   14. Applied egg-rr 90.0%

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

       1. unpow2 90.0%

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

       2. associate-/l/ 90.0%

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

       3. *-commutative 90.0%

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

       4. associate-/l/ 90.0%

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

       5. *-commutative 90.0%

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

   16. Applied egg-rr 90.0%

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

       1. unpow2 90.0%

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

   18. Simplified 90.0%

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

Alternative 3: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 105000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_2}}{t\_3}\right)}^{2}}{t\_m \cdot \left(t\_2 \cdot t\_3\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0)) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 105000000000.0)
      (pow
       (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/ (pow (/ (/ (/ (sqrt 2.0) k_m) t_2) t_3) 2.0) (* t_m (* t_2 t_3)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), -2.0);
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 105000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = pow((((sqrt(2.0) / k_m) / t_2) / t_3), 2.0) / (t_m * (t_2 * t_3));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0);
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 105000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.pow((((Math.sqrt(2.0) / k_m) / t_2) / t_3), 2.0) / (t_m * (t_2 * t_3));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ -2.0
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 105000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_2) / t_3) ^ 2.0) / Float64(t_m * Float64(t_2 * t_3)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 105000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e11

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.05e11 < k

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.6%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.2%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r/ 76.2%

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

   8. Simplified 76.2%

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

      1. frac-times 73.2%

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

      2. associate-/r/ 73.2%

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

      3. associate-/l* 73.3%

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

      4. div-inv 73.3%

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

      5. pow-flip 73.3%

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

      6. metadata-eval 73.3%

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

   10. Applied egg-rr 73.2%

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

       1. times-frac 76.2%

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

       2. associate-*r/ 76.2%

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

       3. associate-/r/ 76.2%

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

       4. associate-/r* 76.2%

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

       5. associate-*l/ 73.2%

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

   12. Simplified 90.0%

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

Alternative 4: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 11200000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\sqrt{2}}{k\_m}}{t\_2}\right)}^{2}}{t\_m \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (pow (cbrt l) -2.0) (cbrt (* (sin k_m) (tan k_m))))))
   (*
    t_s
    (if (<= k_m 11200000000000.0)
      (pow
       (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/ (pow (/ (/ (sqrt 2.0) k_m) t_2) 2.0) (* t_m t_2))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), -2.0) * cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = pow(((sqrt(2.0) / k_m) / t_2), 2.0) / (t_m * t_2);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / t_2), 2.0) / (t_m * t_2);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k_m) * tan(k_m))))
	tmp = 0.0
	if (k_m <= 11200000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(sqrt(2.0) / k_m) / t_2) ^ 2.0) / Float64(t_m * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 11200000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e13

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.12e13 < k

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.6%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.2%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r/ 76.2%

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

   8. Simplified 76.2%

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

      1. frac-times 73.2%

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

      2. associate-/r/ 73.2%

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

      3. associate-/l* 73.3%

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

      4. div-inv 73.3%

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

      5. pow-flip 73.3%

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

      6. metadata-eval 73.3%

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

   10. Applied egg-rr 73.2%

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

       1. times-frac 76.2%

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

       2. associate-*r/ 76.2%

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

       3. associate-/r/ 76.2%

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

       4. associate-/r* 76.2%

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

       5. associate-*l/ 73.2%

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

   12. Simplified 90.0%

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

       1. div-inv 88.6%

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

       2. associate-/l/ 88.6%

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

       3. *-commutative 88.6%

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

   14. Applied egg-rr 88.6%

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

       1. associate-*r/ 89.9%

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

       2. *-rgt-identity 89.9%

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

   16. Simplified 89.9%

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

Alternative 5: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 11200000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_3}}{t\_2}\right)}^{2}}{t\_m}}{t\_2 \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0)) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 11200000000000.0)
      (pow
       (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/ (/ (pow (/ (/ (/ (sqrt 2.0) k_m) t_3) t_2) 2.0) t_m) (* t_2 t_3))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), -2.0);
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (pow((((sqrt(2.0) / k_m) / t_3) / t_2), 2.0) / t_m) / (t_2 * t_3);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0);
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (Math.pow((((Math.sqrt(2.0) / k_m) / t_3) / t_2), 2.0) / t_m) / (t_2 * t_3);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ -2.0
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 11200000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_3) / t_2) ^ 2.0) / t_m) / Float64(t_2 * t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 11200000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e13

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.12e13 < k

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.6%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.2%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r/ 76.2%

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

   8. Simplified 76.2%

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

      1. frac-times 73.2%

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

      2. associate-/r/ 73.2%

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

      3. associate-/l* 73.3%

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

      4. div-inv 73.3%

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

      5. pow-flip 73.3%

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

      6. metadata-eval 73.3%

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

   10. Applied egg-rr 73.2%

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

       1. times-frac 76.2%

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

       2. associate-*r/ 76.2%

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

       3. associate-/r/ 76.2%

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

       4. associate-/r* 76.2%

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

       5. associate-*l/ 73.2%

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

   12. Simplified 90.0%

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

       1. *-commutative 90.0%

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

       2. cbrt-prod 90.0%

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

   14. Applied egg-rr 90.0%

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

       1. div-inv 88.7%

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

       2. associate-/l/ 88.7%

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

       3. cbrt-prod 88.6%

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

       4. *-commutative 88.6%

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

       5. *-commutative 88.6%

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

       6. *-commutative 88.6%

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

       7. cbrt-prod 88.7%

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

   16. Applied egg-rr 88.6%

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

       1. associate-*r/ 89.9%

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

       2. *-rgt-identity 89.9%

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

       3. associate-/r* 88.6%

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

       4. *-commutative 88.6%

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

       5. associate-/r* 88.7%

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

   18. Simplified 88.7%

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

Alternative 6: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 114000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{t\_2}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 6 \cdot 10^{+160}:\\ \;\;\;\;\left(2 \cdot \left(t\_2 \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k\_m}{t\_m}\right)}^{-2}}}{\sqrt[3]{\sin k\_m \cdot \tan k\_m} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 114000000000.0)
      (pow (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt t_2)) 2.0)
      (if (<= k_m 6e+160)
        (* (* 2.0 (* t_2 (pow (* k_m (sin k_m)) -2.0))) (* l l))
        (pow
         (/
          (cbrt (* 2.0 (pow (/ k_m t_m) -2.0)))
          (* (cbrt (* (sin k_m) (tan k_m))) (* t_m (pow (cbrt l) -2.0))))
         3.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 114000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt(t_2)), 2.0);
	} else if (k_m <= 6e+160) {
		tmp = (2.0 * (t_2 * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	} else {
		tmp = pow((cbrt((2.0 * pow((k_m / t_m), -2.0))) / (cbrt((sin(k_m) * tan(k_m))) * (t_m * pow(cbrt(l), -2.0)))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 114000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt(t_2)), 2.0);
	} else if (k_m <= 6e+160) {
		tmp = (2.0 * (t_2 * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	} else {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((k_m / t_m), -2.0))) / (Math.cbrt((Math.sin(k_m) * Math.tan(k_m))) * (t_m * Math.pow(Math.cbrt(l), -2.0)))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 114000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(t_2)) ^ 2.0;
	elseif (k_m <= 6e+160)
		tmp = Float64(Float64(2.0 * Float64(t_2 * (Float64(k_m * sin(k_m)) ^ -2.0))) * Float64(l * l));
	else
		tmp = Float64(cbrt(Float64(2.0 * (Float64(k_m / t_m) ^ -2.0))) / Float64(cbrt(Float64(sin(k_m) * tan(k_m))) * Float64(t_m * (cbrt(l) ^ -2.0)))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 114000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 6e+160], N[(N[(2.0 * N[(t$95$2 * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(k$95$m / t$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.14e11

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.14e11 < k < 5.9999999999999997e160

   1. Initial program 27.6%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in t around 0 85.7%

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

      1. associate-*r/ 85.7%

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

      2. associate-*r* 85.8%

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

      3. *-commutative 85.8%

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

   7. Simplified 85.8%

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

      1. associate-/l* 85.8%

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

      2. associate-*r* 85.8%

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

      3. pow-prod-down 85.7%

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

   9. Applied egg-rr 85.7%

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

       1. *-commutative 85.7%

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

       2. associate-/r* 86.0%

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

       3. *-commutative 86.0%

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

   11. Simplified 86.0%

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

       1. div-inv 86.0%

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

       2. pow-flip 86.0%

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

       3. metadata-eval 86.0%

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

   13. Applied egg-rr 86.0%

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

   if 5.9999999999999997e160 < k

   1. Initial program 29.3%

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

      1. *-commutative 29.3%

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

      2. associate-/r* 29.3%

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

   3. Simplified 29.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 29.3%

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

      2. pow3 29.3%

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

      3. cbrt-prod 29.3%

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

      4. cbrt-div 29.3%

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

      5. rem-cbrt-cube 46.2%

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

      6. cbrt-prod 58.2%

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

      7. pow2 58.2%

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

   6. Applied egg-rr 58.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 58.0%

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

      2. pow3 58.0%

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

   8. Applied egg-rr 63.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 114000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+160}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k}{t} \cdot {\left(k \cdot \sin k\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 11200000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{t\_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_2 \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 11200000000000.0)
      (pow (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt t_2)) 2.0)
      (* (* 2.0 (* t_2 (pow (* k_m (sin k_m)) -2.0))) (* l l))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt(t_2)), 2.0);
	} else {
		tmp = (2.0 * (t_2 * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 11200000000000.0d0) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / sin(k_m))) * sqrt(t_2)) ** 2.0d0
    else
        tmp = (2.0d0 * (t_2 * ((k_m * sin(k_m)) ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 11200000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt(t_2)), 2.0);
	} else {
		tmp = (2.0 * (t_2 * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 11200000000000.0:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / math.sin(k_m))) * math.sqrt(t_2)), 2.0)
	else:
		tmp = (2.0 * (t_2 * math.pow((k_m * math.sin(k_m)), -2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 11200000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(t_2)) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(t_2 * (Float64(k_m * sin(k_m)) ^ -2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 11200000000000.0)
		tmp = (((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt(t_2)) ^ 2.0;
	else
		tmp = (2.0 * (t_2 * ((k_m * sin(k_m)) ^ -2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 11200000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(t$95$2 * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e13

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in k around inf 48.9%

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

      1. times-frac 50.8%

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

   9. Simplified 50.8%

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

   if 1.12e13 < k

   1. Initial program 28.2%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in t around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. associate-*r* 71.1%

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

      3. *-commutative 71.1%

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

   7. Simplified 71.1%

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

      1. associate-/l* 71.1%

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

      2. associate-*r* 71.1%

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

      3. pow-prod-down 71.1%

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

   9. Applied egg-rr 71.1%

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

       1. *-commutative 71.1%

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

       2. associate-/r* 71.3%

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

       3. *-commutative 71.3%

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

   11. Simplified 71.3%

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

       1. div-inv 71.2%

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

       2. pow-flip 71.3%

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

       3. metadata-eval 71.3%

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

   13. Applied egg-rr 71.3%

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

Alternative 8: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sin k\_m\\ t_3 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 105000:\\ \;\;\;\;{\left(\sqrt{t\_3} \cdot \left(\ell \cdot \frac{\sqrt{2}}{t\_2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_3 \cdot {t\_2}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sin k_m))) (t_3 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 105000.0)
      (pow (* (sqrt t_3) (* l (/ (sqrt 2.0) t_2))) 2.0)
      (* (* 2.0 (* t_3 (pow t_2 -2.0))) (* l l))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sin(k_m);
	double t_3 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 105000.0) {
		tmp = pow((sqrt(t_3) * (l * (sqrt(2.0) / t_2))), 2.0);
	} else {
		tmp = (2.0 * (t_3 * pow(t_2, -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = k_m * sin(k_m)
    t_3 = cos(k_m) / t_m
    if (k_m <= 105000.0d0) then
        tmp = (sqrt(t_3) * (l * (sqrt(2.0d0) / t_2))) ** 2.0d0
    else
        tmp = (2.0d0 * (t_3 * (t_2 ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = k_m * Math.sin(k_m);
	double t_3 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 105000.0) {
		tmp = Math.pow((Math.sqrt(t_3) * (l * (Math.sqrt(2.0) / t_2))), 2.0);
	} else {
		tmp = (2.0 * (t_3 * Math.pow(t_2, -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = k_m * math.sin(k_m)
	t_3 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 105000.0:
		tmp = math.pow((math.sqrt(t_3) * (l * (math.sqrt(2.0) / t_2))), 2.0)
	else:
		tmp = (2.0 * (t_3 * math.pow(t_2, -2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sin(k_m))
	t_3 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 105000.0)
		tmp = Float64(sqrt(t_3) * Float64(l * Float64(sqrt(2.0) / t_2))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(t_3 * (t_2 ^ -2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = k_m * sin(k_m);
	t_3 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 105000.0)
		tmp = (sqrt(t_3) * (l * (sqrt(2.0) / t_2))) ^ 2.0;
	else
		tmp = (2.0 * (t_3 * (t_2 ^ -2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 105000.0], N[Power[N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(t$95$3 * N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 105000

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/r* 38.0%

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

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.4%

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

      2. pow2 28.4%

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

   6. Applied egg-rr 31.5%

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

   7. Taylor expanded in k around inf 49.1%

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

      1. associate-/l* 49.1%

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

   9. Simplified 49.1%

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

   if 105000 < k

   1. Initial program 27.8%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. associate-*r* 71.5%

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

      3. *-commutative 71.5%

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

   7. Simplified 71.5%

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

      1. associate-/l* 71.5%

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

      2. associate-*r* 71.5%

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

      3. pow-prod-down 71.5%

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

   9. Applied egg-rr 71.5%

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

       1. *-commutative 71.5%

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

       2. associate-/r* 71.7%

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

       3. *-commutative 71.7%

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

   11. Simplified 71.7%

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

       1. div-inv 71.6%

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

       2. pow-flip 71.7%

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

       3. metadata-eval 71.7%

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

   13. Applied egg-rr 71.7%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 105000:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k}{t} \cdot {\left(k \cdot \sin k\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k\_m}{t\_m} \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.2e-6)
    (pow (* (/ (* l (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (* (* 2.0 (* (/ (cos k_m) t_m) (pow (* k_m (sin k_m)) -2.0))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-6) {
		tmp = pow((((l * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / t_m) * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.2d-6) then
        tmp = (((l * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 7.2e-6) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / t_m) * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 7.2e-6:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / t_m) * math.pow((k_m * math.sin(k_m)), -2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-6)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / t_m) * (Float64(k_m * sin(k_m)) ^ -2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.2e-6)
		tmp = (((l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ^ -2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.2e-6], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999967e-6

   1. Initial program 38.4%

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

      1. *-commutative 38.4%

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

      2. associate-/r* 38.4%

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

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   6. Applied egg-rr 30.7%

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

   7. Taylor expanded in k around 0 41.2%

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

   if 7.19999999999999967e-6 < k

   1. Initial program 27.0%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. associate-*r* 72.4%

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

      3. *-commutative 72.4%

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

   7. Simplified 72.4%

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

      1. associate-/l* 72.4%

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

      2. associate-*r* 72.4%

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

      3. pow-prod-down 72.3%

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

   9. Applied egg-rr 72.3%

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

       1. *-commutative 72.3%

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

       2. associate-/r* 72.5%

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

       3. *-commutative 72.5%

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

   11. Simplified 72.5%

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

       1. div-inv 72.5%

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

       2. pow-flip 72.5%

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

       3. metadata-eval 72.5%

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

   13. Applied egg-rr 72.5%

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

Alternative 10: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t\_m}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k\_m}{t\_m} \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.8e-6)
    (pow (* (sqrt (/ 1.0 t_m)) (* l (/ (sqrt 2.0) (pow k_m 2.0)))) 2.0)
    (* (* 2.0 (* (/ (cos k_m) t_m) (pow (* k_m (sin k_m)) -2.0))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.8e-6) {
		tmp = pow((sqrt((1.0 / t_m)) * (l * (sqrt(2.0) / pow(k_m, 2.0)))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / t_m) * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.8d-6) then
        tmp = (sqrt((1.0d0 / t_m)) * (l * (sqrt(2.0d0) / (k_m ** 2.0d0)))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 6.8e-6) {
		tmp = Math.pow((Math.sqrt((1.0 / t_m)) * (l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0)))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / t_m) * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 6.8e-6:
		tmp = math.pow((math.sqrt((1.0 / t_m)) * (l * (math.sqrt(2.0) / math.pow(k_m, 2.0)))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / t_m) * math.pow((k_m * math.sin(k_m)), -2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 6.8e-6)
		tmp = Float64(sqrt(Float64(1.0 / t_m)) * Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / t_m) * (Float64(k_m * sin(k_m)) ^ -2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.8e-6)
		tmp = (sqrt((1.0 / t_m)) * (l * (sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ^ -2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.8e-6], N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000012e-6

   1. Initial program 38.4%

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

      1. *-commutative 38.4%

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

      2. associate-/r* 38.4%

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

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   6. Applied egg-rr 30.7%

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

   7. Taylor expanded in k around 0 41.2%

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

      1. associate-/l* 41.2%

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

   9. Simplified 41.2%

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

   if 6.80000000000000012e-6 < k

   1. Initial program 27.0%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. associate-*r* 72.4%

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

      3. *-commutative 72.4%

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

   7. Simplified 72.4%

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

      1. associate-/l* 72.4%

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

      2. associate-*r* 72.4%

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

      3. pow-prod-down 72.3%

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

   9. Applied egg-rr 72.3%

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

       1. *-commutative 72.3%

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

       2. associate-/r* 72.5%

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

       3. *-commutative 72.5%

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

   11. Simplified 72.5%

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

       1. div-inv 72.5%

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

       2. pow-flip 72.5%

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

       3. metadata-eval 72.5%

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

   13. Applied egg-rr 72.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k}{t} \cdot {\left(k \cdot \sin k\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{1}{t\_m}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k\_m}{t\_m} \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.5e-6)
    (pow (* l (/ (* (sqrt 2.0) (sqrt (/ 1.0 t_m))) (pow k_m 2.0))) 2.0)
    (* (* 2.0 (* (/ (cos k_m) t_m) (pow (* k_m (sin k_m)) -2.0))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.5e-6) {
		tmp = pow((l * ((sqrt(2.0) * sqrt((1.0 / t_m))) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / t_m) * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.5d-6) then
        tmp = (l * ((sqrt(2.0d0) * sqrt((1.0d0 / t_m))) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 6.5e-6) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) * Math.sqrt((1.0 / t_m))) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / t_m) * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 6.5e-6:
		tmp = math.pow((l * ((math.sqrt(2.0) * math.sqrt((1.0 / t_m))) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / t_m) * math.pow((k_m * math.sin(k_m)), -2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 6.5e-6)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / t_m))) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / t_m) * (Float64(k_m * sin(k_m)) ^ -2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.5e-6)
		tmp = (l * ((sqrt(2.0) * sqrt((1.0 / t_m))) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ^ -2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.5e-6], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.4999999999999996e-6

   1. Initial program 38.4%

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

      1. *-commutative 38.4%

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

      2. associate-/r* 38.4%

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

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   6. Applied egg-rr 30.7%

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

   7. Taylor expanded in k around 0 31.2%

      \[\leadsto {\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k \cdot \left(1 + 0.08333333333333333 \cdot {k}^{2}\right)\right)}}\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 31.2%

         \[\leadsto {\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \left(1 + \color{blue}{{k}^{2} \cdot 0.08333333333333333}\right)\right)}\right)}^{2} \]

   9. Simplified 31.2%

      \[\leadsto {\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k \cdot \left(1 + {k}^{2} \cdot 0.08333333333333333\right)\right)}}\right)}^{2} \]

   10. Taylor expanded in k around 0 41.2%

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

       1. associate-*l/ 39.6%

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

       2. associate-*l* 39.6%

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

       3. associate-/l* 41.1%

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

   12. Simplified 41.1%

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

   if 6.4999999999999996e-6 < k

   1. Initial program 27.0%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. associate-*r* 72.4%

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

      3. *-commutative 72.4%

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

   7. Simplified 72.4%

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

      1. associate-/l* 72.4%

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

      2. associate-*r* 72.4%

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

      3. pow-prod-down 72.3%

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

   9. Applied egg-rr 72.3%

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

       1. *-commutative 72.3%

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

       2. associate-/r* 72.5%

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

       3. *-commutative 72.5%

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

   11. Simplified 72.5%

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

       1. div-inv 72.5%

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

       2. pow-flip 72.5%

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

       3. metadata-eval 72.5%

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

   13. Applied egg-rr 72.5%

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

Alternative 12: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-183}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\sin k\_m}^{2} \cdot t\_2}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+132}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k\_m}{t\_m}}}{k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow k_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.4e-183)
      (* (* l l) (/ 2.0 (* (pow (sin k_m) 2.0) t_2)))
      (if (<= t_m 6e+132)
        (pow (/ (/ (sqrt 2.0) (/ k_m t_m)) (* k_m (/ (pow t_m 1.5) l))) 2.0)
        (* (* l l) (/ 2.0 (* (pow k_m 2.0) t_2))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = t_m * pow(k_m, 2.0);
	double tmp;
	if (t_m <= 4.4e-183) {
		tmp = (l * l) * (2.0 / (pow(sin(k_m), 2.0) * t_2));
	} else if (t_m <= 6e+132) {
		tmp = pow(((sqrt(2.0) / (k_m / t_m)) / (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * t_2));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k_m ** 2.0d0)
    if (t_m <= 4.4d-183) then
        tmp = (l * l) * (2.0d0 / ((sin(k_m) ** 2.0d0) * t_2))
    else if (t_m <= 6d+132) then
        tmp = ((sqrt(2.0d0) / (k_m / t_m)) / (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * t_2))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = t_m * Math.pow(k_m, 2.0);
	double tmp;
	if (t_m <= 4.4e-183) {
		tmp = (l * l) * (2.0 / (Math.pow(Math.sin(k_m), 2.0) * t_2));
	} else if (t_m <= 6e+132) {
		tmp = Math.pow(((Math.sqrt(2.0) / (k_m / t_m)) / (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * t_2));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = t_m * math.pow(k_m, 2.0)
	tmp = 0
	if t_m <= 4.4e-183:
		tmp = (l * l) * (2.0 / (math.pow(math.sin(k_m), 2.0) * t_2))
	elif t_m <= 6e+132:
		tmp = math.pow(((math.sqrt(2.0) / (k_m / t_m)) / (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * t_2))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(t_m * (k_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.4e-183)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((sin(k_m) ^ 2.0) * t_2)));
	elseif (t_m <= 6e+132)
		tmp = Float64(Float64(sqrt(2.0) / Float64(k_m / t_m)) / Float64(k_m * Float64((t_m ^ 1.5) / l))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = t_m * (k_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 4.4e-183)
		tmp = (l * l) * (2.0 / ((sin(k_m) ^ 2.0) * t_2));
	elseif (t_m <= 6e+132)
		tmp = ((sqrt(2.0) / (k_m / t_m)) / (k_m * ((t_m ^ 1.5) / l))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-183], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+132], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.3999999999999999e-183

   1. Initial program 33.6%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in t around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. associate-*r* 70.3%

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

      3. *-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in k around 0 61.2%

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

   if 4.3999999999999999e-183 < t < 5.9999999999999996e132

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-/r* 53.9%

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

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.3%

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

      2. pow2 53.3%

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

   6. Applied egg-rr 65.0%

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

   7. Taylor expanded in k around 0 69.4%

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

   if 5.9999999999999996e132 < t

   1. Initial program 10.8%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in t around 0 76.0%

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

      1. associate-*r/ 76.0%

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

      2. associate-*r* 76.0%

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

      3. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in k around 0 73.5%

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

   9. Taylor expanded in k around 0 73.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-183}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+132}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{k \cdot \frac{{t}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k\_m}{t\_m}}}{k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k\_m}{t\_m} \cdot {\left(k\_m \cdot \sin k\_m\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (pow (/ (/ (sqrt 2.0) (/ k_m t_m)) (* k_m (/ (pow t_m 1.5) l))) 2.0)
    (* (* 2.0 (* (/ (cos k_m) t_m) (pow (* k_m (sin k_m)) -2.0))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = pow(((sqrt(2.0) / (k_m / t_m)) / (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / t_m) * pow((k_m * sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = ((sqrt(2.0d0) / (k_m / t_m)) / (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) / t_m) * ((k_m * sin(k_m)) ** (-2.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = Math.pow(((Math.sqrt(2.0) / (k_m / t_m)) / (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / t_m) * Math.pow((k_m * Math.sin(k_m)), -2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = math.pow(((math.sqrt(2.0) / (k_m / t_m)) / (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / t_m) * math.pow((k_m * math.sin(k_m)), -2.0))) * (l * l)
	return t_s * tmp

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 15.8%

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

      1. *-commutative 15.8%

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

      2. associate-/r* 15.8%

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

   3. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.0%

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

      2. pow2 26.0%

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

   6. Applied egg-rr 23.1%

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

   7. Taylor expanded in k around 0 34.8%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 42.8%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in t around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*r* 79.1%

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

      3. *-commutative 79.1%

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

   7. Simplified 79.1%

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

      1. associate-/l* 79.1%

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

      2. associate-*r* 76.0%

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

      3. pow-prod-down 76.0%

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

   9. Applied egg-rr 76.0%

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

       1. *-commutative 76.0%

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

       2. associate-/r* 76.1%

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

       3. *-commutative 76.1%

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

   11. Simplified 76.1%

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

       1. div-inv 76.1%

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

       2. pow-flip 76.3%

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

       3. metadata-eval 76.3%

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

   13. Applied egg-rr 76.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{k \cdot \frac{{t}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\cos k}{t} \cdot {\left(k \cdot \sin k\right)}^{-2}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * ((2.0 / Math.pow(k_m, 2.0)) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r/ 71.8%

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

   2. associate-*r* 71.8%

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

   3. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in k around 0 63.0%

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

9. Taylor expanded in k around inf 63.0%

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

    1. associate-/r* 63.0%

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

11. Simplified 63.0%

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

12. Final simplification 63.0%

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

Alternative 15: 64.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * (2.0 / (Math.pow(Math.sin(k_m), 2.0) * (t_m * Math.pow(k_m, 2.0)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r/ 71.8%

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

   2. associate-*r* 71.8%

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

   3. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in k around 0 63.0%

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

9. Final simplification 63.0%

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

Alternative 16: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.019:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot {k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t\_m}}{{\left(k\_m \cdot \sin k\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.019)
    (* (* l l) (/ 2.0 (* (pow k_m 2.0) (* t_m (pow k_m 2.0)))))
    (* (* l l) (/ (/ 2.0 t_m) (pow (* k_m (sin k_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.019) {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * (t_m * pow(k_m, 2.0))));
	} else {
		tmp = (l * l) * ((2.0 / t_m) / pow((k_m * sin(k_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.019d0) then
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * (t_m * (k_m ** 2.0d0))))
    else
        tmp = (l * l) * ((2.0d0 / t_m) / ((k_m * sin(k_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 0.019) {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * (t_m * Math.pow(k_m, 2.0))));
	} else {
		tmp = (l * l) * ((2.0 / t_m) / Math.pow((k_m * Math.sin(k_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.019:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * (t_m * math.pow(k_m, 2.0))))
	else:
		tmp = (l * l) * ((2.0 / t_m) / math.pow((k_m * math.sin(k_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.019)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * (k_m ^ 2.0)))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / t_m) / (Float64(k_m * sin(k_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.019)
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * (t_m * (k_m ^ 2.0))));
	else
		tmp = (l * l) * ((2.0 / t_m) / ((k_m * sin(k_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.019], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0189999999999999995

   1. Initial program 38.2%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. associate-*r* 71.8%

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

      3. *-commutative 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in k around 0 66.4%

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

   9. Taylor expanded in k around 0 65.0%

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

   if 0.0189999999999999995 < k

   1. Initial program 27.4%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in t around 0 72.0%

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

      1. associate-*r/ 72.0%

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

      2. associate-*r* 72.0%

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

      3. *-commutative 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in k around 0 53.1%

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

      1. div-inv 53.1%

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

      2. associate-*r* 53.1%

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

      3. pow-prod-down 53.1%

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

   10. Applied egg-rr 53.1%

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

       1. associate-*r/ 53.1%

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

       2. metadata-eval 53.1%

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

       3. *-commutative 53.1%

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

       4. associate-/r* 53.1%

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

       5. *-commutative 53.1%

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

   12. Simplified 53.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.019:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * ((2.0 / t_m) / Math.pow((k_m * Math.sin(k_m)), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r/ 71.8%

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

   2. associate-*r* 71.8%

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

   3. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in k around 0 63.0%

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

   1. div-inv 63.0%

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

   2. associate-*r* 59.9%

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

   3. pow-prod-down 59.9%

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

10. Applied egg-rr 59.9%

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

    1. associate-*r/ 59.9%

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

    2. metadata-eval 59.9%

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

    3. *-commutative 59.9%

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

    4. associate-/r* 59.9%

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

    5. *-commutative 59.9%

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

12. Simplified 59.9%

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

13. Final simplification 59.9%

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

Alternative 18: 63.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * (2.0 * ((Math.cos(k_m) / t_m) / Math.pow(k_m, 4.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r/ 71.8%

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

   2. associate-*r* 71.8%

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

   3. *-commutative 71.8%

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

7. Simplified 71.8%

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

   1. associate-/l* 71.8%

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

   2. associate-*r* 68.7%

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

   3. pow-prod-down 68.8%

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

9. Applied egg-rr 68.8%

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

    1. *-commutative 68.8%

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

    2. associate-/r* 68.8%

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

    3. *-commutative 68.8%

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

11. Simplified 68.8%

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

12. Taylor expanded in k around 0 57.9%

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

13. Final simplification 57.9%

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

Alternative 19: 63.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k\_m}^{-4}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{t\_m}\right) \cdot 0.006944444444444444\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+303)
    (* (* l l) (/ (* 2.0 (pow k_m -4.0)) t_m))
    (* (* 2.0 (/ (* l l) t_m)) 0.006944444444444444))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+303) {
		tmp = (l * l) * ((2.0 * pow(k_m, -4.0)) / t_m);
	} else {
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+303) then
        tmp = (l * l) * ((2.0d0 * (k_m ** (-4.0d0))) / t_m)
    else
        tmp = (2.0d0 * ((l * l) / t_m)) * 0.006944444444444444d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if ((l * l) <= 5e+303) {
		tmp = (l * l) * ((2.0 * Math.pow(k_m, -4.0)) / t_m);
	} else {
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+303:
		tmp = (l * l) * ((2.0 * math.pow(k_m, -4.0)) / t_m)
	else:
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+303)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k_m ^ -4.0)) / t_m));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l * l) / t_m)) * 0.006944444444444444);
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+303)
		tmp = (l * l) * ((2.0 * (k_m ^ -4.0)) / t_m);
	else
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+303], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * 0.006944444444444444), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999997e303

   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 43.5%

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

   5. Taylor expanded in k around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-/r* 60.0%

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

   7. Simplified 60.0%

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

      1. div-inv 60.0%

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

      2. pow-flip 60.0%

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

      3. metadata-eval 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*l/ 60.0%

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

   11. Simplified 60.0%

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

   if 4.9999999999999997e303 < (*.f64 l l)

   1. Initial program 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/r* 35.6%

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

   3. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

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

      2. pow2 19.9%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in k around 0 31.1%

      \[\leadsto {\color{blue}{\left(\frac{-0.08333333333333333 \cdot \left(\left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{t}}\right) + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-*r* 31.1%

         \[\leadsto {\left(\frac{\color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{t}}} + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2} \]

      2. distribute-rgt-out 33.9%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}}{{k}^{2}}\right)}^{2} \]

   9. Simplified 33.9%

      \[\leadsto {\color{blue}{\left(\frac{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}{{k}^{2}}\right)}}^{2} \]

   10. Taylor expanded in k around inf 57.3%

       \[\leadsto \color{blue}{0.006944444444444444 \cdot \frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t}} \]
   11. Step-by-step derivation

       1. *-commutative 57.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t} \cdot 0.006944444444444444} \]

       2. *-commutative 57.3%

          \[\leadsto \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\ell}^{2}}}{t} \cdot 0.006944444444444444 \]

       3. unpow2 57.3%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

       4. rem-square-sqrt 57.3%

          \[\leadsto \frac{\color{blue}{2} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

       5. associate-*r/ 57.3%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right)} \cdot 0.006944444444444444 \]

   12. Simplified 57.3%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right) \cdot 0.006944444444444444} \]
   13. Step-by-step derivation

       1. pow2 57.3%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]

   14. Applied egg-rr 57.3%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{t}\right) \cdot 0.006944444444444444\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{t\_m}\right) \cdot 0.006944444444444444\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+303)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (* (* 2.0 (/ (* l l) t_m)) 0.006944444444444444))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+303) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+303) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (2.0d0 * ((l * l) / t_m)) * 0.006944444444444444d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if ((l * l) <= 5e+303) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+303:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+303)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l * l) / t_m)) * 0.006944444444444444);
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+303)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = (2.0 * ((l * l) / t_m)) * 0.006944444444444444;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+303], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * 0.006944444444444444), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999997e303

   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 43.5%

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

   5. Taylor expanded in k around 0 60.0%

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

   if 4.9999999999999997e303 < (*.f64 l l)

   1. Initial program 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/r* 35.6%

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

   3. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

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

      2. pow2 19.9%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in k around 0 31.1%

      \[\leadsto {\color{blue}{\left(\frac{-0.08333333333333333 \cdot \left(\left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{t}}\right) + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-*r* 31.1%

         \[\leadsto {\left(\frac{\color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{t}}} + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2} \]

      2. distribute-rgt-out 33.9%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}}{{k}^{2}}\right)}^{2} \]

   9. Simplified 33.9%

      \[\leadsto {\color{blue}{\left(\frac{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}{{k}^{2}}\right)}}^{2} \]

   10. Taylor expanded in k around inf 57.3%

       \[\leadsto \color{blue}{0.006944444444444444 \cdot \frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t}} \]
   11. Step-by-step derivation

       1. *-commutative 57.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t} \cdot 0.006944444444444444} \]

       2. *-commutative 57.3%

          \[\leadsto \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\ell}^{2}}}{t} \cdot 0.006944444444444444 \]

       3. unpow2 57.3%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

       4. rem-square-sqrt 57.3%

          \[\leadsto \frac{\color{blue}{2} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

       5. associate-*r/ 57.3%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right)} \cdot 0.006944444444444444 \]

   12. Simplified 57.3%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right) \cdot 0.006944444444444444} \]
   13. Step-by-step derivation

       1. pow2 57.3%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]

   14. Applied egg-rr 57.3%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{t}\right) \cdot 0.006944444444444444\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{0.013888888888888888 \cdot {\ell}^{2}}{t\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ (* 0.013888888888888888 (pow l 2.0)) t_m)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((0.013888888888888888 * pow(l, 2.0)) / t_m);
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((0.013888888888888888d0 * (l ** 2.0d0)) / t_m)
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((0.013888888888888888 * Math.pow(l, 2.0)) / t_m);
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((0.013888888888888888 * math.pow(l, 2.0)) / t_m)

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(0.013888888888888888 * (l ^ 2.0)) / t_m))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((0.013888888888888888 * (l ^ 2.0)) / t_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(0.013888888888888888 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

   1. *-commutative 35.4%

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

   2. associate-/r* 35.5%

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

3. Simplified 40.5%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 27.9%

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

   2. pow2 27.9%

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

6. Applied egg-rr 27.7%

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

7. Taylor expanded in k around 0 29.6%

   \[\leadsto {\color{blue}{\left(\frac{-0.08333333333333333 \cdot \left(\left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{t}}\right) + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
8. Step-by-step derivation

   1. associate-*r* 29.6%

      \[\leadsto {\left(\frac{\color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{t}}} + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2} \]

   2. distribute-rgt-out 30.4%

      \[\leadsto {\left(\frac{\color{blue}{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}}{{k}^{2}}\right)}^{2} \]

9. Simplified 30.4%

   \[\leadsto {\color{blue}{\left(\frac{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}{{k}^{2}}\right)}}^{2} \]

10. Taylor expanded in k around inf 36.8%

    \[\leadsto \color{blue}{0.006944444444444444 \cdot \frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t}} \]
11. Step-by-step derivation

    1. *-commutative 36.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t} \cdot 0.006944444444444444} \]

    2. *-commutative 36.8%

       \[\leadsto \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\ell}^{2}}}{t} \cdot 0.006944444444444444 \]

    3. unpow2 36.8%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

    4. rem-square-sqrt 36.8%

       \[\leadsto \frac{\color{blue}{2} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

    5. associate-*r/ 36.8%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right)} \cdot 0.006944444444444444 \]

12. Simplified 36.8%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right) \cdot 0.006944444444444444} \]

13. Taylor expanded in l around 0 36.8%

    \[\leadsto \color{blue}{0.013888888888888888 \cdot \frac{{\ell}^{2}}{t}} \]
14. Step-by-step derivation

    1. associate-*r/ 36.8%

       \[\leadsto \color{blue}{\frac{0.013888888888888888 \cdot {\ell}^{2}}{t}} \]

15. Simplified 36.8%

    \[\leadsto \color{blue}{\frac{0.013888888888888888 \cdot {\ell}^{2}}{t}} \]
16. Add Preprocessing

Alternative 22: 35.4% accurate, 46.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(2 \cdot \frac{\ell \cdot \ell}{t\_m}\right) \cdot 0.006944444444444444\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* 2.0 (/ (* l l) t_m)) 0.006944444444444444)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 * ((l * l) / t_m)) * 0.006944444444444444);
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((2.0d0 * ((l * l) / t_m)) * 0.006944444444444444d0)
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((2.0 * ((l * l) / t_m)) * 0.006944444444444444);
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((2.0 * ((l * l) / t_m)) * 0.006944444444444444)

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(2.0 * Float64(Float64(l * l) / t_m)) * 0.006944444444444444))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((2.0 * ((l * l) / t_m)) * 0.006944444444444444);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

   1. *-commutative 35.4%

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

   2. associate-/r* 35.5%

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

3. Simplified 40.5%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 27.9%

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

   2. pow2 27.9%

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

6. Applied egg-rr 27.7%

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

7. Taylor expanded in k around 0 29.6%

   \[\leadsto {\color{blue}{\left(\frac{-0.08333333333333333 \cdot \left(\left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{t}}\right) + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
8. Step-by-step derivation

   1. associate-*r* 29.6%

      \[\leadsto {\left(\frac{\color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{t}}} + \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2} \]

   2. distribute-rgt-out 30.4%

      \[\leadsto {\left(\frac{\color{blue}{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}}{{k}^{2}}\right)}^{2} \]

9. Simplified 30.4%

   \[\leadsto {\color{blue}{\left(\frac{\sqrt{\frac{1}{t}} \cdot \left(-0.08333333333333333 \cdot \left({k}^{2} \cdot \left(\ell \cdot \sqrt{2}\right)\right) + \ell \cdot \sqrt{2}\right)}{{k}^{2}}\right)}}^{2} \]

10. Taylor expanded in k around inf 36.8%

    \[\leadsto \color{blue}{0.006944444444444444 \cdot \frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t}} \]
11. Step-by-step derivation

    1. *-commutative 36.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{t} \cdot 0.006944444444444444} \]

    2. *-commutative 36.8%

       \[\leadsto \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\ell}^{2}}}{t} \cdot 0.006944444444444444 \]

    3. unpow2 36.8%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

    4. rem-square-sqrt 36.8%

       \[\leadsto \frac{\color{blue}{2} \cdot {\ell}^{2}}{t} \cdot 0.006944444444444444 \]

    5. associate-*r/ 36.8%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right)} \cdot 0.006944444444444444 \]

12. Simplified 36.8%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t}\right) \cdot 0.006944444444444444} \]
13. Step-by-step derivation

    1. pow2 36.8%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]

14. Applied egg-rr 36.8%

    \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \cdot 0.006944444444444444 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))