Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 87.9%

Time: 18.7s

Alternatives: 23

Speedup: 3.8×

• 39.5% 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: 35.6% 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: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.52 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_m \cdot \sqrt{2}}{k\_m}}{{t\_2}^{2}} \cdot \frac{e^{\log 2 \cdot 0.5}}{t\_2 \cdot \frac{k\_m}{t\_m}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt l) 2.0)) (cbrt (* (sin k_m) (tan k_m))))))
   (*
    t_s
    (if (<= k_m 2e-24)
      (*
       2.0
       (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
      (if (<= k_m 1.52e+147)
        (/
         2.0
         (/
          (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
          (* (cos k_m) (pow l 2.0))))
        (*
         (/ (/ (* t_m (sqrt 2.0)) k_m) (pow t_2 2.0))
         (/ (exp (* (log 2.0) 0.5)) (* t_2 (/ k_m 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) {
	double t_2 = (t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 2e-24) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.52e+147) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	} else {
		tmp = (((t_m * sqrt(2.0)) / k_m) / pow(t_2, 2.0)) * (exp((log(2.0) * 0.5)) / (t_2 * (k_m / t_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 = (t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 2e-24) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.52e+147) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	} else {
		tmp = (((t_m * Math.sqrt(2.0)) / k_m) / Math.pow(t_2, 2.0)) * (Math.exp((Math.log(2.0) * 0.5)) / (t_2 * (k_m / t_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 = Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * tan(k_m))))
	tmp = 0.0
	if (k_m <= 2e-24)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 1.52e+147)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(t_m * sqrt(2.0)) / k_m) / (t_2 ^ 2.0)) * Float64(exp(Float64(log(2.0) * 0.5)) / Float64(t_2 * Float64(k_m / t_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[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $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, 2e-24], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.52e+147], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.99999999999999985e-24 < k < 1.51999999999999992e147

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Taylor expanded in t around 0 83.6%

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

   if 1.51999999999999992e147 < k

   1. Initial program 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-/r* 38.1%

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

   3. Simplified 48.0%

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

      1. add-sqr-sqrt 48.0%

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

      2. add-cube-cbrt 48.0%

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

      3. times-frac 48.0%

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

   6. Applied egg-rr 90.1%

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

      1. associate-/r/ 90.0%

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

      2. associate-/l/ 90.0%

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

   8. Simplified 90.0%

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

      1. associate-*l/ 90.1%

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

   10. Applied egg-rr 90.1%

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

       1. pow1/2 90.1%

          \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{{2}^{0.5}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}} \]

       2. pow-to-exp 90.1%

          \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{e^{\log 2 \cdot 0.5}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}} \]

   12. Applied egg-rr 90.1%

       \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{e^{\log 2 \cdot 0.5}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.52 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{e^{\log 2 \cdot 0.5}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_m \cdot \sqrt{2}}{k\_m}}{{t\_2}^{2}} \cdot \frac{\sqrt{2}}{t\_2 \cdot \frac{k\_m}{t\_m}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt l) 2.0)) (cbrt (* (sin k_m) (tan k_m))))))
   (*
    t_s
    (if (<= k_m 3.7e-25)
      (*
       2.0
       (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
      (if (<= k_m 1.8e+147)
        (/
         2.0
         (/
          (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
          (* (cos k_m) (pow l 2.0))))
        (*
         (/ (/ (* t_m (sqrt 2.0)) k_m) (pow t_2 2.0))
         (/ (sqrt 2.0) (* t_2 (/ k_m 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) {
	double t_2 = (t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 3.7e-25) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.8e+147) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	} else {
		tmp = (((t_m * sqrt(2.0)) / k_m) / pow(t_2, 2.0)) * (sqrt(2.0) / (t_2 * (k_m / t_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 = (t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 3.7e-25) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.8e+147) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	} else {
		tmp = (((t_m * Math.sqrt(2.0)) / k_m) / Math.pow(t_2, 2.0)) * (Math.sqrt(2.0) / (t_2 * (k_m / t_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 = Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * tan(k_m))))
	tmp = 0.0
	if (k_m <= 3.7e-25)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 1.8e+147)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(t_m * sqrt(2.0)) / k_m) / (t_2 ^ 2.0)) * Float64(sqrt(2.0) / Float64(t_2 * Float64(k_m / t_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[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $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, 3.7e-25], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.8e+147], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.70000000000000009e-25

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 3.70000000000000009e-25 < k < 1.8000000000000001e147

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Taylor expanded in t around 0 83.6%

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

   if 1.8000000000000001e147 < k

   1. Initial program 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-/r* 38.1%

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

   3. Simplified 48.0%

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

      1. add-sqr-sqrt 48.0%

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

      2. add-cube-cbrt 48.0%

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

      3. times-frac 48.0%

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

   6. Applied egg-rr 90.1%

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

      1. associate-/r/ 90.0%

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

      2. associate-/l/ 90.0%

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

   8. Simplified 90.0%

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

      1. associate-*l/ 90.1%

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

   10. Applied egg-rr 90.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\sqrt{2}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.58 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{t\_2 \cdot \frac{k\_m}{t\_m}} \cdot \frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{{t\_2}^{2}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt l) 2.0)) (cbrt (* (sin k_m) (tan k_m))))))
   (*
    t_s
    (if (<= k_m 1.58e-24)
      (*
       2.0
       (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
      (if (<= k_m 1.35e+147)
        (/
         2.0
         (/
          (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
          (* (cos k_m) (pow l 2.0))))
        (*
         (/ (sqrt 2.0) (* t_2 (/ k_m t_m)))
         (/ (* t_m (/ (sqrt 2.0) k_m)) (pow t_2 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 t_2 = (t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 1.58e-24) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.35e+147) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	} else {
		tmp = (sqrt(2.0) / (t_2 * (k_m / t_m))) * ((t_m * (sqrt(2.0) / k_m)) / pow(t_2, 2.0));
	}
	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 = (t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 1.58e-24) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.35e+147) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	} else {
		tmp = (Math.sqrt(2.0) / (t_2 * (k_m / t_m))) * ((t_m * (Math.sqrt(2.0) / k_m)) / Math.pow(t_2, 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)
	t_2 = Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * tan(k_m))))
	tmp = 0.0
	if (k_m <= 1.58e-24)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 1.35e+147)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(t_2 * Float64(k_m / t_m))) * Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / (t_2 ^ 2.0)));
	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[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $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, 1.58e-24], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.35e+147], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5799999999999999e-24

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.5799999999999999e-24 < k < 1.34999999999999999e147

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Taylor expanded in t around 0 83.6%

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

   if 1.34999999999999999e147 < k

   1. Initial program 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-/r* 38.1%

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

   3. Simplified 48.0%

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

      1. add-sqr-sqrt 48.0%

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

      2. add-cube-cbrt 48.0%

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

      3. times-frac 48.0%

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

   6. Applied egg-rr 90.1%

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

      1. associate-/r/ 90.0%

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

      2. associate-/l/ 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}} \cdot \frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{{\left(t\_m \cdot \left(t\_2 \cdot t\_3\right)\right)}^{2} \cdot \frac{\left(k\_m \cdot t\_2\right) \cdot \left(t\_m \cdot t\_3\right)}{t\_m}}\\ \end{array} \end{array} \end{array} \]

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 3.7e-25)
      (*
       2.0
       (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
      (if (<= k_m 4.9e+148)
        (/
         2.0
         (/
          (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
          (* (cos k_m) (pow l 2.0))))
        (*
         (sqrt 2.0)
         (/
          (* t_m (/ (sqrt 2.0) k_m))
          (*
           (pow (* t_m (* t_2 t_3)) 2.0)
           (/ (* (* k_m t_2) (* t_m t_3)) 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) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 3.7e-25) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 4.9e+148) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	} else {
		tmp = sqrt(2.0) * ((t_m * (sqrt(2.0) / k_m)) / (pow((t_m * (t_2 * t_3)), 2.0) * (((k_m * t_2) * (t_m * t_3)) / t_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.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 3.7e-25) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 4.9e+148) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	} else {
		tmp = Math.sqrt(2.0) * ((t_m * (Math.sqrt(2.0) / k_m)) / (Math.pow((t_m * (t_2 * t_3)), 2.0) * (((k_m * t_2) * (t_m * t_3)) / t_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(Float64(sin(k_m) * tan(k_m)))
	t_3 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 3.7e-25)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 4.9e+148)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / Float64((Float64(t_m * Float64(t_2 * t_3)) ^ 2.0) * Float64(Float64(Float64(k_m * t_2) * Float64(t_m * t_3)) / t_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[(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, 3.7e-25], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.9e+148], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$m * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k$95$m * t$95$2), $MachinePrecision] * N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.70000000000000009e-25

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 3.70000000000000009e-25 < k < 4.9e148

   1. Initial program 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in t around 0 82.0%

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

   if 4.9e148 < k

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/r* 42.1%

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

   3. Simplified 47.4%

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

      1. add-sqr-sqrt 47.4%

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

      2. add-cube-cbrt 47.4%

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

      3. times-frac 47.4%

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

   6. Applied egg-rr 89.2%

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

      1. associate-/r/ 89.1%

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

      2. associate-/l/ 89.1%

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

   8. Simplified 89.1%

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

      1. associate-*l/ 89.2%

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

   10. Applied egg-rr 89.2%

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

       1. frac-times 89.0%

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

       2. associate-/l* 89.0%

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

       3. div-inv 89.0%

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

       4. pow-flip 89.0%

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

       5. metadata-eval 89.0%

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

       6. associate-*l* 88.9%

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

   12. Applied egg-rr 88.9%

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

       1. *-commutative 88.9%

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

       2. associate-/l* 89.1%

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

       3. associate-*r/ 89.0%

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

       4. *-commutative 89.0%

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

       5. associate-/l* 89.0%

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

       6. associate-*l* 89.2%

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

   14. Simplified 89.3%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2} \cdot \frac{\left(k \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.52 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

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 2e-24)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (if (<= k_m 1.52e+147)
      (/
       2.0
       (/
        (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
        (* (cos k_m) (pow l 2.0))))
      (/
       2.0
       (pow
        (*
         (/ t_m (pow (cbrt l) 2.0))
         (cbrt (* (sin k_m) (* (tan k_m) (pow (/ k_m t_m) 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 tmp;
	if (k_m <= 2e-24) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.52e+147) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * (tan(k_m) * pow((k_m / t_m), 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 tmp;
	if (k_m <= 2e-24) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.52e+147) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * Math.pow((k_m / t_m), 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)
	tmp = 0.0
	if (k_m <= 2e-24)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 1.52e+147)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * (Float64(k_m / t_m) ^ 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_] := N[(t$95$s * If[LessEqual[k$95$m, 2e-24], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.52e+147], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.99999999999999985e-24 < k < 1.51999999999999992e147

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Taylor expanded in t around 0 83.6%

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

   if 1.51999999999999992e147 < k

   1. Initial program 38.1%

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

   3. Simplified 38.1%

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

      1. add-cube-cbrt 38.1%

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

      2. pow3 38.1%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.52 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

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 2e-24)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (/
     2.0
     (/
      (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
      (* (cos k_m) (pow l 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 <= 2e-24) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 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 <= 2d-24) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t_m * (sin(k_m) ** 2.0d0))) / (cos(k_m) * (l ** 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 <= 2e-24) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 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 <= 2e-24:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t_m * math.pow(math.sin(k_m), 2.0))) / (math.cos(k_m) * math.pow(l, 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 <= 2e-24)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 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 <= 2e-24)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t_m * (sin(k_m) ^ 2.0))) / (cos(k_m) * (l ^ 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, 2e-24], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.99999999999999985e-24 < k

   1. Initial program 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in t around 0 77.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k\_m}^{2}} \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

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 8.2e-25)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     (/ 2.0 (* t_m (pow k_m 2.0)))
     (/ (* (cos k_m) (pow l 2.0)) (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) {
	double tmp;
	if (k_m <= 8.2e-25) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = (2.0 / (t_m * pow(k_m, 2.0))) * ((cos(k_m) * pow(l, 2.0)) / pow(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 <= 8.2d-25) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = (2.0d0 / (t_m * (k_m ** 2.0d0))) * ((cos(k_m) * (l ** 2.0d0)) / (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 <= 8.2e-25) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = (2.0 / (t_m * Math.pow(k_m, 2.0))) * ((Math.cos(k_m) * Math.pow(l, 2.0)) / Math.pow(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 <= 8.2e-25:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = (2.0 / (t_m * math.pow(k_m, 2.0))) * ((math.cos(k_m) * math.pow(l, 2.0)) / math.pow(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 <= 8.2e-25)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) * Float64(Float64(cos(k_m) * (l ^ 2.0)) / (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 <= 8.2e-25)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = (2.0 / (t_m * (k_m ^ 2.0))) * ((cos(k_m) * (l ^ 2.0)) / (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, 8.2e-25], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.19999999999999974e-25

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 8.19999999999999974e-25 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. associate-*r* 77.7%

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

      3. times-frac 77.7%

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

      4. *-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-204}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{1}{\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \left(k\_m \cdot \sin k\_m\right)}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= (* l l) 1e-204)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (*
     2.0
     (pow (* l (/ 1.0 (* (sqrt (/ t_m (cos k_m))) (* 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 ((l * l) <= 1e-204) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * pow((l * (1.0 / (sqrt((t_m / cos(k_m))) * (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 ((l * l) <= 1d-204) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((l * (1.0d0 / (sqrt((t_m / cos(k_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 ((l * l) <= 1e-204) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * Math.pow((l * (1.0 / (Math.sqrt((t_m / Math.cos(k_m))) * (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 (l * l) <= 1e-204:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * math.pow((l * (1.0 / (math.sqrt((t_m / math.cos(k_m))) * (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 (Float64(l * l) <= 1e-204)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(l * Float64(1.0 / Float64(sqrt(Float64(t_m / cos(k_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 ((l * l) <= 1e-204)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((l * (1.0 / (sqrt((t_m / cos(k_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[N[(l * l), $MachinePrecision], 1e-204], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(l * N[(1.0 / N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-204

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

      2. associate-/r* 22.4%

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

   3. Simplified 29.4%

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

   6. Applied egg-rr 23.3%

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

   7. Taylor expanded in k around 0 42.5%

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

   if 1e-204 < (*.f64 l l)

   1. Initial program 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-/r* 32.4%

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

   3. Simplified 38.4%

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

   6. Applied egg-rr 29.1%

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

   7. Taylor expanded in t around 0 52.1%

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

      1. associate-/l* 52.1%

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

   9. Simplified 52.1%

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

       1. div-inv 52.1%

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

       2. unpow-prod-down 52.0%

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

       3. pow2 52.0%

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

       4. pow1/2 52.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 52.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 52.1%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 52.1%

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

       8. metadata-eval 52.1%

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

   11. Applied egg-rr 52.1%

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

       1. associate-*r/ 52.1%

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

       2. associate-*l/ 51.6%

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

       3. associate-/r/ 51.6%

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

   13. Simplified 51.6%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-204}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{1}{\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \sin k\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot {k\_m}^{2}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

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 3.7e-25)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     (/ (* 2.0 (cos k_m)) (* (pow (sin k_m) 2.0) (* t_m (pow 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 <= 3.7e-25) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = ((2.0 * cos(k_m)) / (pow(sin(k_m), 2.0) * (t_m * pow(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 <= 3.7d-25) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = ((2.0d0 * cos(k_m)) / ((sin(k_m) ** 2.0d0) * (t_m * (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 <= 3.7e-25) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k_m)) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * Math.pow(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 <= 3.7e-25:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k_m)) / (math.pow(math.sin(k_m), 2.0) * (t_m * math.pow(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 <= 3.7e-25)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * (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 <= 3.7e-25)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = ((2.0 * cos(k_m)) / ((sin(k_m) ^ 2.0) * (t_m * (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, 3.7e-25], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / 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] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.70000000000000009e-25

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 3.70000000000000009e-25 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. associate-*r/ 77.6%

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

      2. associate-*r* 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m}}\\ \end{array} \end{array} \]

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 1e-25)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     (* l l)
     (/ 2.0 (/ (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0))) (cos 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 tmp;
	if (k_m <= 1e-25) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / cos(k_m)));
	}
	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 <= 1d-25) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 / (((k_m ** 2.0d0) * (t_m * (sin(k_m) ** 2.0d0))) / cos(k_m)))
    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 <= 1e-25) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / Math.cos(k_m)));
	}
	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 <= 1e-25:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = (l * l) * (2.0 / ((math.pow(k_m, 2.0) * (t_m * math.pow(math.sin(k_m), 2.0))) / math.cos(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)
	tmp = 0.0
	if (k_m <= 1e-25)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / cos(k_m))));
	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 <= 1e-25)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = (l * l) * (2.0 / (((k_m ^ 2.0) * (t_m * (sin(k_m) ^ 2.0))) / cos(k_m)));
	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, 1e-25], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000004e-25

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.00000000000000004e-25 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 2e-24)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     (* l l)
     (* 2.0 (/ (/ (cos k_m) (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) {
	double tmp;
	if (k_m <= 2e-24) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k_m) / pow(k_m, 2.0)) / (t_m * pow(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 <= 2d-24) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) / (t_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 <= 2e-24) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) / (t_m * Math.pow(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 <= 2e-24:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) / (t_m * math.pow(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 <= 2e-24)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / Float64(t_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 <= 2e-24)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = (l * l) * (2.0 * ((cos(k_m) / (k_m ^ 2.0)) / (t_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, 2e-24], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.2%

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

   6. Applied egg-rr 28.3%

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

   7. Taylor expanded in t around 0 48.6%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.0%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.0%

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

       8. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

   if 1.99999999999999985e-24 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. associate-/r* 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{k\_m}^{2}} \cdot \frac{\ell \cdot \ell}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 5e-11)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     2.0
     (*
      (/ (cos k_m) (pow k_m 2.0))
      (/ (* l l) (* 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) {
	double tmp;
	if (k_m <= 5e-11) {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(k_m, 2.0)) * ((l * l) / (t_m * pow(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 <= 5d-11) then
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) * ((l * l) / (t_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 <= 5e-11) {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) * ((l * l) / (t_m * Math.pow(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 <= 5e-11:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) * ((l * l) / (t_m * math.pow(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 <= 5e-11)
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) * Float64(Float64(l * l) / Float64(t_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 <= 5e-11)
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (k_m ^ 2.0)) * ((l * l) / (t_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, 5e-11], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000018e-11

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-/r* 29.0%

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

   3. Simplified 34.5%

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

   6. Applied egg-rr 28.7%

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

   7. Taylor expanded in t around 0 48.7%

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

      1. associate-/l* 49.0%

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

   9. Simplified 49.0%

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

       1. div-inv 49.0%

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

       2. unpow-prod-down 49.0%

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

       3. pow2 49.0%

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

       4. pow1/2 49.0%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 49.0%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 49.1%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 49.1%

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

       8. metadata-eval 49.1%

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

   11. Applied egg-rr 49.1%

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

   if 5.00000000000000018e-11 < k

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.1%

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

      3. times-frac 38.1%

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

   6. Applied egg-rr 82.1%

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

      1. associate-/r/ 82.1%

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

      2. associate-/l/ 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in k around inf 75.6%

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

       1. associate-*r* 75.7%

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

       2. unpow2 75.7%

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

       3. rem-square-sqrt 76.0%

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

       4. associate-*l/ 76.0%

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

       5. *-commutative 76.0%

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

       6. *-commutative 76.0%

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

       7. times-frac 72.5%

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

       8. *-commutative 72.5%

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

   11. Simplified 72.5%

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

       1. pow2 72.5%

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

   13. Applied egg-rr 72.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-102}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= l 1.9e-102)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (*
     2.0
     (pow (/ 1.0 (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos 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 (l <= 1.9e-102) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * pow((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(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 (l <= 1.9d-102) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((1.0d0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(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 (l <= 1.9e-102) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * Math.pow((1.0 / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(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 l <= 1.9e-102:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * math.pow((1.0 / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(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 (l <= 1.9e-102)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(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 (l <= 1.9e-102)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((1.0 / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(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[l, 1.9e-102], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(1.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.90000000000000013e-102

   1. Initial program 28.3%

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

      1. *-commutative 28.3%

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

      2. associate-/r* 28.3%

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

   3. Simplified 36.1%

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

   6. Applied egg-rr 28.7%

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

   7. Taylor expanded in k around 0 43.3%

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

   if 1.90000000000000013e-102 < l

   1. Initial program 29.9%

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

      1. *-commutative 29.9%

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

      2. associate-/r* 29.9%

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

   3. Simplified 33.5%

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

   6. Applied egg-rr 24.0%

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

   7. Taylor expanded in t around 0 45.2%

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

      1. associate-/l* 45.2%

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

   9. Simplified 45.2%

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

       1. div-inv 45.2%

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

       2. unpow-prod-down 45.1%

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

       3. pow2 45.1%

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

       4. pow1/2 45.1%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       5. pow1/2 45.1%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       6. pow-prod-up 45.2%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2} \]

       7. metadata-eval 45.2%

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

       8. metadata-eval 45.2%

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

   11. Applied egg-rr 45.2%

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

Alternative 14: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+215}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{k\_m}^{2}} \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 6e+215)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (*
     2.0
     (* (/ (cos k_m) (pow k_m 2.0)) (/ (/ (pow l 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) {
	double tmp;
	if (l <= 6e+215) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(k_m, 2.0)) * ((pow(l, 2.0) / t_m) / pow(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 (l <= 6d+215) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) * (((l ** 2.0d0) / t_m) / (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 (l <= 6e+215) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) * ((Math.pow(l, 2.0) / t_m) / Math.pow(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 l <= 6e+215:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) * ((math.pow(l, 2.0) / t_m) / math.pow(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 (l <= 6e+215)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) * Float64(Float64((l ^ 2.0) / t_m) / (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 (l <= 6e+215)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((cos(k_m) / (k_m ^ 2.0)) * (((l ^ 2.0) / t_m) / (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[l, 6e+215], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.9999999999999998e215

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

      2. associate-/r* 27.9%

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

   3. Simplified 34.8%

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

   6. Applied egg-rr 26.4%

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

   7. Taylor expanded in k around 0 37.3%

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

   if 5.9999999999999998e215 < l

   1. Initial program 40.0%

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

      1. *-commutative 40.0%

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

      2. associate-/r* 40.0%

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

   3. Simplified 40.0%

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

      1. add-sqr-sqrt 40.0%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 80.1%

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

      1. associate-/r/ 80.1%

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

      2. associate-/l/ 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in k around inf 90.5%

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

       1. associate-*r* 90.5%

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

       2. unpow2 90.5%

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

       3. rem-square-sqrt 90.5%

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

       4. associate-*l/ 90.5%

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

       5. *-commutative 90.5%

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

       6. *-commutative 90.5%

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

       7. times-frac 90.5%

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

       8. *-commutative 90.5%

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

   11. Simplified 90.5%

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

   12. Taylor expanded in k around 0 90.5%

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

       1. *-commutative 90.5%

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

       2. associate-/r* 90.5%

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

   14. Simplified 90.5%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+215}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 19500:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 8 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({k\_m}^{-2} \cdot \left(\ell \cdot \sqrt{\frac{2}{t\_m}}\right)\right)}^{2}\\ \end{array} \end{array} \]

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 19500.0)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (if (<= k_m 8e+150)
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l) (/ t_m l)))
        (/ (/ k_m t_m) (/ t_m k_m))))
      (pow (* (pow k_m -2.0) (* l (sqrt (/ 2.0 t_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 <= 19500.0) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else if (k_m <= 8e+150) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = pow((pow(k_m, -2.0) * (l * sqrt((2.0 / t_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 <= 19500.0d0) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else if (k_m <= 8d+150) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)))
    else
        tmp = ((k_m ** (-2.0d0)) * (l * sqrt((2.0d0 / t_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 <= 19500.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else if (k_m <= 8e+150) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = Math.pow((Math.pow(k_m, -2.0) * (l * Math.sqrt((2.0 / t_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 <= 19500.0:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	elif k_m <= 8e+150:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)))
	else:
		tmp = math.pow((math.pow(k_m, -2.0) * (l * math.sqrt((2.0 / t_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 <= 19500.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	elseif (k_m <= 8e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(Float64(k_m / t_m) / Float64(t_m / k_m))));
	else
		tmp = Float64((k_m ^ -2.0) * Float64(l * sqrt(Float64(2.0 / t_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 <= 19500.0)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	elseif (k_m <= 8e+150)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	else
		tmp = ((k_m ^ -2.0) * (l * sqrt((2.0 / t_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, 19500.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 8e+150], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 19500

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-/r* 29.1%

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

   3. Simplified 34.4%

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

   6. Applied egg-rr 29.3%

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

   7. Taylor expanded in k around 0 39.5%

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

   if 19500 < k < 7.99999999999999985e150

   1. Initial program 17.6%

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

   3. Simplified 17.7%

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

      1. unpow3 17.7%

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

      2. times-frac 25.1%

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

      3. pow2 25.1%

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

   6. Applied egg-rr 25.1%

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

      1. +-commutative 25.1%

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

      2. associate-+l- 48.9%

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

      3. metadata-eval 48.9%

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

      4. --rgt-identity 48.9%

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

      5. unpow2 48.9%

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

      6. clear-num 48.8%

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

      7. un-div-inv 48.9%

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

   8. Applied egg-rr 48.9%

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

   if 7.99999999999999985e150 < k

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/r* 44.4%

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

   3. Simplified 50.0%

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

   6. Applied egg-rr 22.4%

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

   7. Taylor expanded in k around 0 50.8%

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

      1. associate-*l/ 50.8%

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

      2. associate-*l* 50.8%

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

      3. associate-/l* 50.8%

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

   9. Simplified 50.8%

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

       1. pow1 50.8%

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

       2. div-inv 50.8%

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

       3. sqrt-unprod 50.8%

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

       4. div-inv 50.8%

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

       5. pow-flip 50.8%

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

       6. metadata-eval 50.8%

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

   11. Applied egg-rr 50.8%

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

       1. unpow1 50.8%

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

       2. associate-*r* 50.8%

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

       3. *-commutative 50.8%

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

   13. Simplified 50.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 19500:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({k}^{-2} \cdot \left(\ell \cdot \sqrt{\frac{2}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 25000:\\ \;\;\;\;{\left(\ell \cdot \frac{t\_2}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 4.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({k\_m}^{-2} \cdot \left(\ell \cdot t\_2\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ 2.0 t_m))))
   (*
    t_s
    (if (<= k_m 25000.0)
      (pow (* l (/ t_2 (pow k_m 2.0))) 2.0)
      (if (<= k_m 4.5e+150)
        (/
         2.0
         (*
          (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (/ (/ k_m t_m) (/ t_m k_m))))
        (pow (* (pow k_m -2.0) (* l t_2)) 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 t_2 = sqrt((2.0 / t_m));
	double tmp;
	if (k_m <= 25000.0) {
		tmp = pow((l * (t_2 / pow(k_m, 2.0))), 2.0);
	} else if (k_m <= 4.5e+150) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = pow((pow(k_m, -2.0) * (l * t_2)), 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) :: t_2
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / t_m))
    if (k_m <= 25000.0d0) then
        tmp = (l * (t_2 / (k_m ** 2.0d0))) ** 2.0d0
    else if (k_m <= 4.5d+150) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)))
    else
        tmp = ((k_m ** (-2.0d0)) * (l * t_2)) ** 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 t_2 = Math.sqrt((2.0 / t_m));
	double tmp;
	if (k_m <= 25000.0) {
		tmp = Math.pow((l * (t_2 / Math.pow(k_m, 2.0))), 2.0);
	} else if (k_m <= 4.5e+150) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = Math.pow((Math.pow(k_m, -2.0) * (l * t_2)), 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):
	t_2 = math.sqrt((2.0 / t_m))
	tmp = 0
	if k_m <= 25000.0:
		tmp = math.pow((l * (t_2 / math.pow(k_m, 2.0))), 2.0)
	elif k_m <= 4.5e+150:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)))
	else:
		tmp = math.pow((math.pow(k_m, -2.0) * (l * t_2)), 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)
	t_2 = sqrt(Float64(2.0 / t_m))
	tmp = 0.0
	if (k_m <= 25000.0)
		tmp = Float64(l * Float64(t_2 / (k_m ^ 2.0))) ^ 2.0;
	elseif (k_m <= 4.5e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(Float64(k_m / t_m) / Float64(t_m / k_m))));
	else
		tmp = Float64((k_m ^ -2.0) * Float64(l * t_2)) ^ 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)
	t_2 = sqrt((2.0 / t_m));
	tmp = 0.0;
	if (k_m <= 25000.0)
		tmp = (l * (t_2 / (k_m ^ 2.0))) ^ 2.0;
	elseif (k_m <= 4.5e+150)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l) * (t_m / l))) * ((k_m / t_m) / (t_m / k_m)));
	else
		tmp = ((k_m ^ -2.0) * (l * t_2)) ^ 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_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 25000.0], N[Power[N[(l * N[(t$95$2 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 4.5e+150], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 25000

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-/r* 29.1%

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

   3. Simplified 34.4%

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

   6. Applied egg-rr 29.3%

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

   7. Taylor expanded in k around 0 39.5%

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

      1. associate-*l/ 39.1%

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

      2. associate-*l* 39.1%

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

      3. associate-/l* 39.5%

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

   9. Simplified 39.5%

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

       1. associate-*r/ 39.1%

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

       2. sqrt-unprod 39.1%

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

       3. div-inv 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. associate-/l* 39.5%

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

   13. Simplified 39.5%

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

   if 25000 < k < 4.5e150

   1. Initial program 17.6%

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

   3. Simplified 17.7%

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

      1. unpow3 17.7%

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

      2. times-frac 25.1%

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

      3. pow2 25.1%

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

   6. Applied egg-rr 25.1%

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

      1. +-commutative 25.1%

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

      2. associate-+l- 48.9%

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

      3. metadata-eval 48.9%

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

      4. --rgt-identity 48.9%

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

      5. unpow2 48.9%

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

      6. clear-num 48.8%

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

      7. un-div-inv 48.9%

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

   8. Applied egg-rr 48.9%

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

   if 4.5e150 < k

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/r* 44.4%

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

   3. Simplified 50.0%

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

   6. Applied egg-rr 22.4%

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

   7. Taylor expanded in k around 0 50.8%

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

      1. associate-*l/ 50.8%

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

      2. associate-*l* 50.8%

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

      3. associate-/l* 50.8%

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

   9. Simplified 50.8%

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

       1. pow1 50.8%

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

       2. div-inv 50.8%

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

       3. sqrt-unprod 50.8%

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

       4. div-inv 50.8%

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

       5. pow-flip 50.8%

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

       6. metadata-eval 50.8%

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

   11. Applied egg-rr 50.8%

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

       1. unpow1 50.8%

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

       2. associate-*r* 50.8%

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

       3. *-commutative 50.8%

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

   13. Simplified 50.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 25000:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({k}^{-2} \cdot \left(\ell \cdot \sqrt{\frac{2}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 25000:\\ \;\;\;\;{\left(\ell \cdot \frac{t\_2}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({k\_m}^{-2} \cdot \left(\ell \cdot t\_2\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ 2.0 t_m))))
   (*
    t_s
    (if (<= k_m 25000.0)
      (pow (* l (/ t_2 (pow k_m 2.0))) 2.0)
      (if (<= k_m 5.2e+150)
        (/
         2.0
         (*
          (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (* (/ k_m t_m) (/ k_m t_m))))
        (pow (* (pow k_m -2.0) (* l t_2)) 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 t_2 = sqrt((2.0 / t_m));
	double tmp;
	if (k_m <= 25000.0) {
		tmp = pow((l * (t_2 / pow(k_m, 2.0))), 2.0);
	} else if (k_m <= 5.2e+150) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) * (k_m / t_m)));
	} else {
		tmp = pow((pow(k_m, -2.0) * (l * t_2)), 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) :: t_2
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / t_m))
    if (k_m <= 25000.0d0) then
        tmp = (l * (t_2 / (k_m ** 2.0d0))) ** 2.0d0
    else if (k_m <= 5.2d+150) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l) * (t_m / l))) * ((k_m / t_m) * (k_m / t_m)))
    else
        tmp = ((k_m ** (-2.0d0)) * (l * t_2)) ** 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 t_2 = Math.sqrt((2.0 / t_m));
	double tmp;
	if (k_m <= 25000.0) {
		tmp = Math.pow((l * (t_2 / Math.pow(k_m, 2.0))), 2.0);
	} else if (k_m <= 5.2e+150) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) * (k_m / t_m)));
	} else {
		tmp = Math.pow((Math.pow(k_m, -2.0) * (l * t_2)), 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):
	t_2 = math.sqrt((2.0 / t_m))
	tmp = 0
	if k_m <= 25000.0:
		tmp = math.pow((l * (t_2 / math.pow(k_m, 2.0))), 2.0)
	elif k_m <= 5.2e+150:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))) * ((k_m / t_m) * (k_m / t_m)))
	else:
		tmp = math.pow((math.pow(k_m, -2.0) * (l * t_2)), 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)
	t_2 = sqrt(Float64(2.0 / t_m))
	tmp = 0.0
	if (k_m <= 25000.0)
		tmp = Float64(l * Float64(t_2 / (k_m ^ 2.0))) ^ 2.0;
	elseif (k_m <= 5.2e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(Float64(k_m / t_m) * Float64(k_m / t_m))));
	else
		tmp = Float64((k_m ^ -2.0) * Float64(l * t_2)) ^ 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)
	t_2 = sqrt((2.0 / t_m));
	tmp = 0.0;
	if (k_m <= 25000.0)
		tmp = (l * (t_2 / (k_m ^ 2.0))) ^ 2.0;
	elseif (k_m <= 5.2e+150)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l) * (t_m / l))) * ((k_m / t_m) * (k_m / t_m)));
	else
		tmp = ((k_m ^ -2.0) * (l * t_2)) ^ 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_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 25000.0], N[Power[N[(l * N[(t$95$2 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 5.2e+150], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 25000

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-/r* 29.1%

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

   3. Simplified 34.4%

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

   6. Applied egg-rr 29.3%

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

   7. Taylor expanded in k around 0 39.5%

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

      1. associate-*l/ 39.1%

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

      2. associate-*l* 39.1%

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

      3. associate-/l* 39.5%

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

   9. Simplified 39.5%

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

       1. associate-*r/ 39.1%

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

       2. sqrt-unprod 39.1%

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

       3. div-inv 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. associate-/l* 39.5%

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

   13. Simplified 39.5%

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

   if 25000 < k < 5.20000000000000012e150

   1. Initial program 17.6%

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

   3. Simplified 17.7%

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

      1. unpow3 17.7%

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

      2. times-frac 25.1%

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

      3. pow2 25.1%

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

   6. Applied egg-rr 25.1%

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

      1. +-commutative 25.1%

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

      2. associate-+l- 48.9%

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

      3. metadata-eval 48.9%

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

      4. --rgt-identity 48.9%

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

      5. unpow2 48.9%

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

   8. Applied egg-rr 48.9%

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

   if 5.20000000000000012e150 < k

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/r* 44.4%

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

   3. Simplified 50.0%

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

   6. Applied egg-rr 22.4%

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

   7. Taylor expanded in k around 0 50.8%

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

      1. associate-*l/ 50.8%

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

      2. associate-*l* 50.8%

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

      3. associate-/l* 50.8%

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

   9. Simplified 50.8%

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

       1. pow1 50.8%

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

       2. div-inv 50.8%

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

       3. sqrt-unprod 50.8%

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

       4. div-inv 50.8%

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

       5. pow-flip 50.8%

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

       6. metadata-eval 50.8%

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

   11. Applied egg-rr 50.8%

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

       1. unpow1 50.8%

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

       2. associate-*r* 50.8%

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

       3. *-commutative 50.8%

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

   13. Simplified 50.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 25000:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({k}^{-2} \cdot \left(\ell \cdot \sqrt{\frac{2}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.4% accurate, 1.4× speedup

Math

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

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 (pow (* l (/ (sqrt (/ 2.0 t_m)) (pow k_m 2.0))) 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 * pow((l * (sqrt((2.0 / t_m)) / pow(k_m, 2.0))), 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 * (sqrt((2.0d0 / t_m)) / (k_m ** 2.0d0))) ** 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 * Math.pow((l * (Math.sqrt((2.0 / t_m)) / Math.pow(k_m, 2.0))), 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 * math.pow((l * (math.sqrt((2.0 / t_m)) / math.pow(k_m, 2.0))), 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(l * Float64(sqrt(Float64(2.0 / t_m)) / (k_m ^ 2.0))) ^ 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 * (sqrt((2.0 / t_m)) / (k_m ^ 2.0))) ^ 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[Power[N[(l * N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/r* 28.9%

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

3. Simplified 35.2%

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

6. Applied egg-rr 27.1%

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

7. Taylor expanded in k around 0 37.2%

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

   1. associate-*l/ 36.9%

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

   2. associate-*l* 36.9%

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

   3. associate-/l* 37.2%

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

9. Simplified 37.2%

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

    1. associate-*r/ 36.9%

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

    2. sqrt-unprod 36.9%

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

    3. div-inv 36.9%

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

11. Applied egg-rr 36.9%

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

    1. associate-/l* 37.2%

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

13. Simplified 37.2%

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

Alternative 19: 73.2% accurate, 1.4× speedup

Math

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

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 (pow (* l (* (pow k_m -2.0) (sqrt (/ 2.0 t_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 * pow((l * (pow(k_m, -2.0) * sqrt((2.0 / t_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 * ((k_m ** (-2.0d0)) * sqrt((2.0d0 / t_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 * Math.pow((l * (Math.pow(k_m, -2.0) * Math.sqrt((2.0 / t_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 * math.pow((l * (math.pow(k_m, -2.0) * math.sqrt((2.0 / t_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(l * Float64((k_m ^ -2.0) * sqrt(Float64(2.0 / t_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 * ((k_m ^ -2.0) * sqrt((2.0 / t_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[Power[N[(l * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/r* 28.9%

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

3. Simplified 35.2%

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

6. Applied egg-rr 27.1%

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

7. Taylor expanded in k around 0 37.2%

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

   1. associate-*l/ 36.9%

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

   2. associate-*l* 36.9%

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

   3. associate-/l* 37.2%

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

9. Simplified 37.2%

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

    1. pow1 37.2%

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

    2. div-inv 37.1%

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

    3. sqrt-unprod 37.1%

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

    4. div-inv 37.1%

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

    5. pow-flip 37.1%

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

    6. metadata-eval 37.1%

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

11. Applied egg-rr 37.1%

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

    1. unpow1 37.1%

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

13. Simplified 37.1%

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

14. Final simplification 37.1%

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

Alternative 20: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{4} \cdot \left(t\_m + \left(t\_m \cdot {k\_m}^{2}\right) \cdot 0.16666666666666666\right)}\right) \end{array} \]

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 k_m 4.0) (+ t_m (* (* t_m (pow k_m 2.0)) 0.16666666666666666)))))))

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, 4.0) * (t_m + ((t_m * pow(k_m, 2.0)) * 0.16666666666666666)))));
}

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 ** 4.0d0) * (t_m + ((t_m * (k_m ** 2.0d0)) * 0.16666666666666666d0)))))
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, 4.0) * (t_m + ((t_m * Math.pow(k_m, 2.0)) * 0.16666666666666666)))));
}

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, 4.0) * (t_m + ((t_m * math.pow(k_m, 2.0)) * 0.16666666666666666)))))

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((k_m ^ 4.0) * Float64(t_m + Float64(Float64(t_m * (k_m ^ 2.0)) * 0.16666666666666666))))))
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 ^ 4.0) * (t_m + ((t_m * (k_m ^ 2.0)) * 0.16666666666666666)))));
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[k$95$m, 4.0], $MachinePrecision] * N[(t$95$m + N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 37.6%

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

5. Taylor expanded in k around 0 60.6%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.6%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \]
7. Add Preprocessing

Alternative 21: 62.1% accurate, 2.0× speedup

Math

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

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 (/ (pow l 2.0) (* 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 * (2.0 * (pow(l, 2.0) / (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 * (2.0d0 * ((l ** 2.0d0) / (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 * (2.0 * (Math.pow(l, 2.0) / (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 * (2.0 * (math.pow(l, 2.0) / (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(2.0 * Float64((l ^ 2.0) / Float64(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 * (2.0 * ((l ^ 2.0) / (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[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 37.6%

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

5. Taylor expanded in k around 0 60.6%

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

6. Final simplification 60.6%

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

Alternative 22: 62.1% accurate, 3.8× speedup

Math

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

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 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 / (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 / (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 / (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 / (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(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 / (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[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 37.6%

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

5. Taylor expanded in k around 0 60.6%

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

6. Final simplification 60.6%

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

Alternative 23: 62.1% accurate, 3.8× speedup

Math

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

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 k_m -4.0) 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 * ((l * l) * (2.0 * (pow(k_m, -4.0) / 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 * ((l * l) * (2.0d0 * ((k_m ** (-4.0d0)) / 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 * ((l * l) * (2.0 * (Math.pow(k_m, -4.0) / 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 * ((l * l) * (2.0 * (math.pow(k_m, -4.0) / 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(l * l) * Float64(2.0 * Float64((k_m ^ -4.0) / 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 * ((l * l) * (2.0 * ((k_m ^ -4.0) / 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[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Power[k$95$m, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 37.6%

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

5. Taylor expanded in k around 0 60.6%

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

   1. div-inv 60.6%

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

   2. *-commutative 60.6%

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

7. Applied egg-rr 60.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{t \cdot {k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. associate-*r/ 60.6%

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

   2. metadata-eval 60.6%

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

   3. associate-/r* 60.6%

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

9. Simplified 60.6%

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

    1. div-inv 60.6%

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

    2. pow-flip 60.6%

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

    3. metadata-eval 60.6%

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

11. Applied egg-rr 60.6%

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

    1. associate-*l/ 60.6%

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

    2. metadata-eval 60.6%

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

    3. pow-sqr 60.6%

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

    4. unpow2 60.6%

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

    5. associate-/l* 60.6%

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

    6. unpow2 60.6%

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

    7. pow-sqr 60.6%

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

    8. metadata-eval 60.6%

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

13. Simplified 60.6%

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

14. Final simplification 60.6%

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

Reproduce

herbie shell --seed 2024137 
(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))))