Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 94.7%

Time: 27.8s

Alternatives: 16

Speedup: 2.0×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 35.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 560000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k_m}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {\sin k_m}^{2}}{\cos k_m \cdot {\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 560000000.0)
    (pow
     (* (/ l (/ (sin k_m) (sqrt 2.0))) (/ (sqrt (/ (cos k_m) t_m)) k_m))
     2.0)
    (/
     2.0
     (/
      (* t_m (pow (sin k_m) 2.0))
      (* (cos k_m) (pow (/ l (fabs 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 <= 560000000.0) {
		tmp = pow(((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((t_m * pow(sin(k_m), 2.0)) / (cos(k_m) * pow((l / fabs(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 <= 560000000.0d0) then
        tmp = ((l / (sin(k_m) / sqrt(2.0d0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / ((t_m * (sin(k_m) ** 2.0d0)) / (cos(k_m) * ((l / abs(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 <= 560000000.0) {
		tmp = Math.pow(((l / (Math.sin(k_m) / Math.sqrt(2.0))) * (Math.sqrt((Math.cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((t_m * Math.pow(Math.sin(k_m), 2.0)) / (Math.cos(k_m) * Math.pow((l / Math.abs(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 <= 560000000.0:
		tmp = math.pow(((l / (math.sin(k_m) / math.sqrt(2.0))) * (math.sqrt((math.cos(k_m) / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 / ((t_m * math.pow(math.sin(k_m), 2.0)) / (math.cos(k_m) * math.pow((l / math.fabs(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 <= 560000000.0)
		tmp = Float64(Float64(l / Float64(sin(k_m) / sqrt(2.0))) * Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / Float64(cos(k_m) * (Float64(l / abs(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 <= 560000000.0)
		tmp = ((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / ((t_m * (sin(k_m) ^ 2.0)) / (cos(k_m) * ((l / abs(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, 560000000.0], N[Power[N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.6e8

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 41.4%

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

      1. associate-*l/ 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 43.4%

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

   10. Simplified 43.4%

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

       1. expm1-log1p-u 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-udef 23.9%

          \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       3. associate-/l* 23.9%

          \[\leadsto {\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\frac{\sin k}{\sqrt{2}}}}\right)} - 1\right) \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

   12. Applied egg-rr 23.9%

       \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-def 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-log1p 43.4%

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

   14. Simplified 43.4%

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

   if 5.6e8 < k

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-*l/ 29.3%

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

      3. associate--l+ 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in t around 0 77.5%

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

      1. times-frac 76.3%

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

   6. Simplified 76.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   10. Simplified 76.3%

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

       1. inv-pow 76.3%

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

       2. add-sqr-sqrt 76.1%

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

       3. unpow-prod-down 76.1%

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

       4. div-inv 76.1%

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

       5. sqrt-prod 76.2%

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

       6. unpow2 76.2%

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

       7. sqrt-prod 39.5%

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

       8. add-sqr-sqrt 62.6%

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

       9. pow-flip 62.6%

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

       10. metadata-eval 62.6%

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

   12. Applied egg-rr 82.5%

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

       1. pow-sqr 82.5%

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

       2. sqr-pow 82.5%

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

       3. rem-sqrt-square 93.9%

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

       4. metadata-eval 93.9%

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

       5. unpow-1 93.9%

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

       6. metadata-eval 93.9%

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

   14. Simplified 93.9%

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

   15. Taylor expanded in l around 0 77.5%

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

       1. unpow2 77.5%

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

       2. unpow-1 77.5%

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

       3. metadata-eval 77.5%

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

       4. unpow-1 77.5%

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

       5. metadata-eval 77.5%

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

       6. sqr-abs 77.5%

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

       7. sqr-pow 77.4%

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

       8. associate-*r* 77.5%

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

       9. *-commutative 77.5%

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

       10. associate-*l* 77.5%

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

       11. unpow2 77.5%

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

       12. sqr-pow 77.5%

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

       13. sqr-abs 77.5%

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

   17. Simplified 93.9%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 560000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\left(\frac{\ell}{\left|k\right|}\right)}^{2}}}\\ \end{array} \]

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k_m}{t_m}\\ t_3 := \ell \cdot \sqrt{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{t_3}{{k_m}^{2}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+252}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k_m}{\sqrt{2}}}\right)}^{2} \cdot \left(t_2 \cdot {k_m}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{t_2} \cdot \frac{t_3}{k_m \cdot \sin k_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)) (t_3 (* l (sqrt 2.0))))
   (*
    t_s
    (if (<= (* l l) 2e-323)
      (pow (* (/ t_3 (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 2e+252)
        (* (pow (/ l (/ (sin k_m) (sqrt 2.0))) 2.0) (* t_2 (pow k_m -2.0)))
        (pow (* (sqrt t_2) (/ t_3 (* 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 t_2 = cos(k_m) / t_m;
	double t_3 = l * sqrt(2.0);
	double tmp;
	if ((l * l) <= 2e-323) {
		tmp = pow(((t_3 / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+252) {
		tmp = pow((l / (sin(k_m) / sqrt(2.0))), 2.0) * (t_2 * pow(k_m, -2.0));
	} else {
		tmp = pow((sqrt(t_2) * (t_3 / (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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    t_3 = l * sqrt(2.0d0)
    if ((l * l) <= 2d-323) then
        tmp = ((t_3 / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 2d+252) then
        tmp = ((l / (sin(k_m) / sqrt(2.0d0))) ** 2.0d0) * (t_2 * (k_m ** (-2.0d0)))
    else
        tmp = (sqrt(t_2) * (t_3 / (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 t_2 = Math.cos(k_m) / t_m;
	double t_3 = l * Math.sqrt(2.0);
	double tmp;
	if ((l * l) <= 2e-323) {
		tmp = Math.pow(((t_3 / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+252) {
		tmp = Math.pow((l / (Math.sin(k_m) / Math.sqrt(2.0))), 2.0) * (t_2 * Math.pow(k_m, -2.0));
	} else {
		tmp = Math.pow((Math.sqrt(t_2) * (t_3 / (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):
	t_2 = math.cos(k_m) / t_m
	t_3 = l * math.sqrt(2.0)
	tmp = 0
	if (l * l) <= 2e-323:
		tmp = math.pow(((t_3 / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 2e+252:
		tmp = math.pow((l / (math.sin(k_m) / math.sqrt(2.0))), 2.0) * (t_2 * math.pow(k_m, -2.0))
	else:
		tmp = math.pow((math.sqrt(t_2) * (t_3 / (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)
	t_2 = Float64(cos(k_m) / t_m)
	t_3 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (Float64(l * l) <= 2e-323)
		tmp = Float64(Float64(t_3 / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 2e+252)
		tmp = Float64((Float64(l / Float64(sin(k_m) / sqrt(2.0))) ^ 2.0) * Float64(t_2 * (k_m ^ -2.0)));
	else
		tmp = Float64(sqrt(t_2) * Float64(t_3 / 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)
	t_2 = cos(k_m) / t_m;
	t_3 = l * sqrt(2.0);
	tmp = 0.0;
	if ((l * l) <= 2e-323)
		tmp = ((t_3 / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 2e+252)
		tmp = ((l / (sin(k_m) / sqrt(2.0))) ^ 2.0) * (t_2 * (k_m ^ -2.0));
	else
		tmp = (sqrt(t_2) * (t_3 / (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_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-323], N[Power[N[(N[(t$95$3 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+252], N[(N[Power[N[(l / N[(N[Sin[k$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$2 * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(t$95$3 / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.97626e-323

   1. Initial program 19.7%

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

      1. associate-/r* 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*l* 19.7%

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

      4. associate-*l/ 19.7%

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

      5. +-commutative 19.7%

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

      6. unpow2 19.7%

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

      7. sqr-neg 19.7%

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

      8. distribute-frac-neg 19.7%

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

      9. distribute-frac-neg 19.7%

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

      10. unpow2 19.7%

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

      11. associate--l+ 30.1%

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

      12. metadata-eval 30.1%

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

      13. +-rgt-identity 30.1%

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

      14. unpow2 30.1%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 30.1%

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

      16. distribute-frac-neg 30.1%

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

   3. Simplified 30.1%

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

      1. add-sqr-sqrt 29.7%

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

   5. Applied egg-rr 11.0%

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

      1. unpow2 11.0%

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

      2. associate-/l/ 11.0%

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

      3. *-commutative 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in k around 0 37.7%

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

   if 1.97626e-323 < (*.f64 l l) < 2.0000000000000002e252

   1. Initial program 42.8%

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

      1. associate-/r* 42.8%

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

      2. *-commutative 42.8%

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

      3. associate-*l* 42.8%

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

      4. associate-*l/ 42.8%

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

      5. +-commutative 42.8%

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

      6. unpow2 42.8%

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

      7. sqr-neg 42.8%

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

      8. distribute-frac-neg 42.8%

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

      9. distribute-frac-neg 42.8%

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

      10. unpow2 42.8%

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

      11. associate--l+ 56.8%

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

      12. metadata-eval 56.8%

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

      13. +-rgt-identity 56.8%

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

      14. unpow2 56.8%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 56.8%

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

      16. distribute-frac-neg 56.8%

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

   3. Simplified 56.8%

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

      1. add-sqr-sqrt 41.8%

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

   5. Applied egg-rr 21.4%

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

      1. unpow2 21.4%

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

      2. associate-/l/ 21.4%

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

      3. *-commutative 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in k around inf 46.8%

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

      1. associate-*l/ 46.8%

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

      2. *-commutative 46.8%

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

      3. times-frac 46.9%

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

   10. Simplified 46.9%

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

       1. unpow-prod-down 44.7%

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

       2. associate-/l* 44.7%

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

       3. div-inv 44.7%

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

       4. unpow-prod-down 44.5%

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

       5. pow2 44.5%

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

       6. add-sqr-sqrt 96.2%

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

       7. pow2 96.2%

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

       8. inv-pow 96.2%

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

       9. inv-pow 96.2%

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

       10. pow-prod-up 96.2%

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

       11. metadata-eval 96.2%

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

   12. Applied egg-rr 96.2%

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

   if 2.0000000000000002e252 < (*.f64 l l)

   1. Initial program 21.2%

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

      1. associate-/r* 21.2%

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

      2. *-commutative 21.2%

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

      3. associate-*l* 21.2%

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

      4. associate-*l/ 21.2%

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

      5. +-commutative 21.2%

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

      6. unpow2 21.2%

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

      7. sqr-neg 21.2%

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

      8. distribute-frac-neg 21.2%

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

      9. distribute-frac-neg 21.2%

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

      10. unpow2 21.2%

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

      11. associate--l+ 22.7%

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

      12. metadata-eval 22.7%

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

      13. +-rgt-identity 22.7%

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

      14. unpow2 22.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 22.7%

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

      16. distribute-frac-neg 22.7%

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

   3. Simplified 22.7%

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

      1. add-sqr-sqrt 7.2%

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

   5. Applied egg-rr 25.3%

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

      1. unpow2 25.3%

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

      2. associate-/l/ 25.4%

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

      3. *-commutative 25.4%

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

   7. Simplified 25.4%

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

   8. Taylor expanded in k around inf 45.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+252}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)}^{2} \cdot \left(\frac{\cos k}{t} \cdot {k}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]

Alternative 3: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 620000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k_m}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{-2} \cdot \frac{t_m \cdot {\sin k_m}^{2}}{\cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 620000000.0)
    (pow
     (* (/ l (/ (sin k_m) (sqrt 2.0))) (/ (sqrt (/ (cos k_m) t_m)) k_m))
     2.0)
    (/
     2.0
     (*
      (pow (/ l (fabs 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 <= 620000000.0) {
		tmp = pow(((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / (pow((l / fabs(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 <= 620000000.0d0) then
        tmp = ((l / (sin(k_m) / sqrt(2.0d0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / (((l / abs(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 <= 620000000.0) {
		tmp = Math.pow(((l / (Math.sin(k_m) / Math.sqrt(2.0))) * (Math.sqrt((Math.cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((l / Math.abs(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 <= 620000000.0:
		tmp = math.pow(((l / (math.sin(k_m) / math.sqrt(2.0))) * (math.sqrt((math.cos(k_m) / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 / (math.pow((l / math.fabs(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 <= 620000000.0)
		tmp = Float64(Float64(l / Float64(sin(k_m) / sqrt(2.0))) * Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64((Float64(l / abs(k_m)) ^ -2.0) * Float64(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 <= 620000000.0)
		tmp = ((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / (((l / abs(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, 620000000.0], N[Power[N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e8

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 41.4%

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

      1. associate-*l/ 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 43.4%

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

   10. Simplified 43.4%

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

       1. expm1-log1p-u 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-udef 23.9%

          \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       3. associate-/l* 23.9%

          \[\leadsto {\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\frac{\sin k}{\sqrt{2}}}}\right)} - 1\right) \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

   12. Applied egg-rr 23.9%

       \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-def 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-log1p 43.4%

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

   14. Simplified 43.4%

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

   if 6.2e8 < k

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-*l/ 29.3%

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

      3. associate--l+ 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in t around 0 77.5%

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

      1. times-frac 76.3%

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

   6. Simplified 76.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   10. Simplified 76.3%

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

       1. inv-pow 76.3%

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

       2. add-sqr-sqrt 76.1%

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

       3. unpow-prod-down 76.1%

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

       4. div-inv 76.1%

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

       5. sqrt-prod 76.2%

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

       6. unpow2 76.2%

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

       7. sqrt-prod 39.5%

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

       8. add-sqr-sqrt 62.6%

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

       9. pow-flip 62.6%

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

       10. metadata-eval 62.6%

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

   12. Applied egg-rr 82.5%

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

       1. pow-sqr 82.5%

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

       2. sqr-pow 82.5%

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

       3. rem-sqrt-square 93.9%

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

       4. metadata-eval 93.9%

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

       5. unpow-1 93.9%

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

       6. metadata-eval 93.9%

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

   14. Simplified 93.9%

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

       1. expm1-log1p-u 92.5%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \left|\frac{1}{k}\right|\right)}^{-2}\right)\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       2. expm1-udef 75.3%

          \[\leadsto \frac{2}{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \left|\frac{1}{k}\right|\right)}^{-2}\right)} - 1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       3. fabs-div 75.3%

          \[\leadsto \frac{2}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \color{blue}{\frac{\left|1\right|}{\left|k\right|}}\right)}^{-2}\right)} - 1\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       4. metadata-eval 75.3%

          \[\leadsto \frac{2}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{\color{blue}{1}}{\left|k\right|}\right)}^{-2}\right)} - 1\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   16. Applied egg-rr 75.3%

       \[\leadsto \frac{2}{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{\left|k\right|}\right)}^{-2}\right)} - 1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]
   17. Step-by-step derivation

       1. expm1-def 92.5%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot \frac{1}{\left|k\right|}\right)}^{-2}\right)\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       2. expm1-log1p 93.9%

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

       3. associate-*r/ 93.9%

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

       4. *-rgt-identity 93.9%

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

   18. Simplified 93.9%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 620000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\ell}{\left|k\right|}\right)}^{-2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 4: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 500000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k_m}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{{\ell}^{2}} \cdot \frac{t_m \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)}{\cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 500000000.0)
    (pow
     (* (/ l (/ (sin k_m) (sqrt 2.0))) (/ (sqrt (/ (cos k_m) t_m)) k_m))
     2.0)
    (/
     2.0
     (*
      (/ (pow k_m 2.0) (pow l 2.0))
      (/ (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 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 <= 500000000.0) {
		tmp = pow(((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) / pow(l, 2.0)) * ((t_m * (0.5 - (cos((k_m * 2.0)) / 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 <= 500000000.0d0) then
        tmp = ((l / (sin(k_m) / sqrt(2.0d0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) / (l ** 2.0d0)) * ((t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 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 <= 500000000.0) {
		tmp = Math.pow(((l / (Math.sin(k_m) / Math.sqrt(2.0))) * (Math.sqrt((Math.cos(k_m) / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) / Math.pow(l, 2.0)) * ((t_m * (0.5 - (Math.cos((k_m * 2.0)) / 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 <= 500000000.0:
		tmp = math.pow(((l / (math.sin(k_m) / math.sqrt(2.0))) * (math.sqrt((math.cos(k_m) / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) / math.pow(l, 2.0)) * ((t_m * (0.5 - (math.cos((k_m * 2.0)) / 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 <= 500000000.0)
		tmp = Float64(Float64(l / Float64(sin(k_m) / sqrt(2.0))) * Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 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 <= 500000000.0)
		tmp = ((l / (sin(k_m) / sqrt(2.0))) * (sqrt((cos(k_m) / t_m)) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / (((k_m ^ 2.0) / (l ^ 2.0)) * ((t_m * (0.5 - (cos((k_m * 2.0)) / 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, 500000000.0], N[Power[N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $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 < 5e8

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 41.4%

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

      1. associate-*l/ 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 43.4%

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

   10. Simplified 43.4%

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

       1. expm1-log1p-u 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-udef 23.9%

          \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{\sin k}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       3. associate-/l* 23.9%

          \[\leadsto {\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\frac{\sin k}{\sqrt{2}}}}\right)} - 1\right) \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

   12. Applied egg-rr 23.9%

       \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)} - 1\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-def 30.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}}\right)\right)} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2} \]

       2. expm1-log1p 43.4%

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

   14. Simplified 43.4%

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

   if 5e8 < k

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-*l/ 29.3%

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

      3. associate--l+ 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in t around 0 77.5%

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

      1. times-frac 76.3%

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

   6. Simplified 76.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. unpow2 76.3%

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

      2. sin-mult 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. div-sub 76.3%

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

      2. +-inverses 76.3%

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

      3. cos-0 76.3%

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

      4. metadata-eval 76.3%

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

      5. count-2 76.3%

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

      6. *-commutative 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 500000000:\\ \;\;\;\;{\left(\frac{\ell}{\frac{\sin k}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 5: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 4000000:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m} \cdot \frac{\ell \cdot \sqrt{2}}{k_m}\right)}^{2}\\ \mathbf{elif}\;k_m \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{\sin k_m \cdot {t_m}^{3}}{\ell \cdot \ell} \cdot t_2}\\ \mathbf{elif}\;k_m \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left({k_m}^{-2} \cdot \frac{{\ell}^{2}}{t_m}\right)} + -1\right)\\ \mathbf{elif}\;k_m \leq 9.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{{t_m}^{3}}{\ell} \cdot \frac{\sin k_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))))
   (*
    t_s
    (if (<= k_m 4000000.0)
      (pow (* (/ (sqrt (/ (cos k_m) t_m)) k_m) (/ (* l (sqrt 2.0)) k_m)) 2.0)
      (if (<= k_m 1.8e+18)
        (/ 2.0 (* (/ (* (sin k_m) (pow t_m 3.0)) (* l l)) t_2))
        (if (<= k_m 1.6e+25)
          (*
           -0.3333333333333333
           (+ (exp (log1p (* (pow k_m -2.0) (/ (pow l 2.0) t_m)))) -1.0))
          (if (<= k_m 9.2e+53)
            (/ 2.0 (* t_2 (* (/ (pow t_m 3.0) l) (/ (sin k_m) l))))
            (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m)))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = tan(k_m) * (k_m / (t_m * (t_m / k_m)));
	double tmp;
	if (k_m <= 4000000.0) {
		tmp = pow(((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0)) / k_m)), 2.0);
	} else if (k_m <= 1.8e+18) {
		tmp = 2.0 / (((sin(k_m) * pow(t_m, 3.0)) / (l * l)) * t_2);
	} else if (k_m <= 1.6e+25) {
		tmp = -0.3333333333333333 * (exp(log1p((pow(k_m, -2.0) * (pow(l, 2.0) / t_m)))) + -1.0);
	} else if (k_m <= 9.2e+53) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 3.0) / l) * (sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / 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.tan(k_m) * (k_m / (t_m * (t_m / k_m)));
	double tmp;
	if (k_m <= 4000000.0) {
		tmp = Math.pow(((Math.sqrt((Math.cos(k_m) / t_m)) / k_m) * ((l * Math.sqrt(2.0)) / k_m)), 2.0);
	} else if (k_m <= 1.8e+18) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.pow(t_m, 3.0)) / (l * l)) * t_2);
	} else if (k_m <= 1.6e+25) {
		tmp = -0.3333333333333333 * (Math.exp(Math.log1p((Math.pow(k_m, -2.0) * (Math.pow(l, 2.0) / t_m)))) + -1.0);
	} else if (k_m <= 9.2e+53) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 3.0) / l) * (Math.sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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):
	t_2 = math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))
	tmp = 0
	if k_m <= 4000000.0:
		tmp = math.pow(((math.sqrt((math.cos(k_m) / t_m)) / k_m) * ((l * math.sqrt(2.0)) / k_m)), 2.0)
	elif k_m <= 1.8e+18:
		tmp = 2.0 / (((math.sin(k_m) * math.pow(t_m, 3.0)) / (l * l)) * t_2)
	elif k_m <= 1.6e+25:
		tmp = -0.3333333333333333 * (math.exp(math.log1p((math.pow(k_m, -2.0) * (math.pow(l, 2.0) / t_m)))) + -1.0)
	elif k_m <= 9.2e+53:
		tmp = 2.0 / (t_2 * ((math.pow(t_m, 3.0) / l) * (math.sin(k_m) / l)))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m))))
	tmp = 0.0
	if (k_m <= 4000000.0)
		tmp = Float64(Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m) * Float64(Float64(l * sqrt(2.0)) / k_m)) ^ 2.0;
	elseif (k_m <= 1.8e+18)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * (t_m ^ 3.0)) / Float64(l * l)) * t_2));
	elseif (k_m <= 1.6e+25)
		tmp = Float64(-0.3333333333333333 * Float64(exp(log1p(Float64((k_m ^ -2.0) * Float64((l ^ 2.0) / t_m)))) + -1.0));
	elseif (k_m <= 9.2e+53)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k_m) / l))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / 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[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4000000.0], N[Power[N[(N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.8e+18], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6e+25], N[(-0.3333333333333333 * N[(N[Exp[N[Log[1 + N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.2e+53], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if k < 4e6

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 41.4%

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

      1. associate-*l/ 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 43.4%

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

   10. Simplified 43.4%

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

   11. Taylor expanded in k around 0 36.1%

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

   if 4e6 < k < 1.8e18

   1. Initial program 2.4%

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

      1. associate-*l* 2.4%

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

      2. associate-*l/ 2.4%

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

      3. associate--l+ 2.4%

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

   3. Simplified 2.4%

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

      1. associate-+r- 2.4%

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

      2. add-exp-log 2.4%

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

      3. log1p-udef 2.4%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 98.8%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 98.8%

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

      6. unpow2 98.8%

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

      7. clear-num 98.8%

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

      8. frac-times 100.0%

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

      9. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 1.8e18 < k < 1.6e25

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. associate-*l/ 0.0%

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

      5. +-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. sqr-neg 0.0%

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

      8. distribute-frac-neg 0.0%

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

      9. distribute-frac-neg 0.0%

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

      10. unpow2 0.0%

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

      11. associate--l+ 0.7%

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

      12. metadata-eval 0.7%

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

      13. +-rgt-identity 0.7%

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

      14. unpow2 0.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 0.7%

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

      16. distribute-frac-neg 0.7%

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

   3. Simplified 0.7%

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

   4. Taylor expanded in k around 0 17.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 50.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 50.7%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 50.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 50.7%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 50.7%

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

      3. metadata-eval 50.7%

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

   9. Applied egg-rr 50.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 17.2%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 17.4%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 17.4%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 17.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]

   if 1.6e25 < k < 9.20000000000000079e53

   1. Initial program 30.7%

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

      1. associate-*l* 30.7%

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

      2. associate-*l/ 30.7%

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

      3. associate--l+ 30.7%

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

   3. Simplified 30.7%

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

      1. associate-+r- 30.7%

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

      2. add-exp-log 28.2%

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

      3. log1p-udef 28.2%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 55.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 58.2%

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

      6. unpow2 58.2%

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

      7. clear-num 57.7%

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

      8. frac-times 57.9%

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

      9. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. times-frac 70.6%

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

   7. Applied egg-rr 70.6%

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

   if 9.20000000000000079e53 < k

   1. Initial program 34.3%

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

      1. associate-/r* 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*l* 34.3%

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

      4. associate-*l/ 34.3%

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

      5. +-commutative 34.3%

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

      6. unpow2 34.3%

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

      7. sqr-neg 34.3%

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

      8. distribute-frac-neg 34.3%

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

      9. distribute-frac-neg 34.3%

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

      10. unpow2 34.3%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 59.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 61.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 61.9%

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

      3. metadata-eval 61.9%

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

   9. Applied egg-rr 61.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 60.9%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 60.9%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 60.9%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 61.9%

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

       3. associate-*r/ 61.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 61.8%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 63.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4000000:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k}{t}}}{k} \cdot \frac{\ell \cdot \sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} + -1\right)\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.5 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m} \cdot \frac{\ell \cdot \sqrt{2}}{k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k_m \cdot \sin k_m\right)}^{2}}{\cos k_m \cdot \frac{{\ell}^{2}}{t_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.5e-28)
    (pow (* (/ (sqrt (/ (cos k_m) t_m)) k_m) (/ (* l (sqrt 2.0)) k_m)) 2.0)
    (/
     2.0
     (/ (pow (* k_m (sin k_m)) 2.0) (* (cos k_m) (/ (pow l 2.0) t_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-28) {
		tmp = pow(((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / (pow((k_m * sin(k_m)), 2.0) / (cos(k_m) * (pow(l, 2.0) / t_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 <= 1.5d-28) then
        tmp = ((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0d0)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / (((k_m * sin(k_m)) ** 2.0d0) / (cos(k_m) * ((l ** 2.0d0) / t_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 <= 1.5e-28) {
		tmp = Math.pow(((Math.sqrt((Math.cos(k_m) / t_m)) / k_m) * ((l * Math.sqrt(2.0)) / k_m)), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((k_m * Math.sin(k_m)), 2.0) / (Math.cos(k_m) * (Math.pow(l, 2.0) / t_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 <= 1.5e-28:
		tmp = math.pow(((math.sqrt((math.cos(k_m) / t_m)) / k_m) * ((l * math.sqrt(2.0)) / k_m)), 2.0)
	else:
		tmp = 2.0 / (math.pow((k_m * math.sin(k_m)), 2.0) / (math.cos(k_m) * (math.pow(l, 2.0) / 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)
	tmp = 0.0
	if (k_m <= 1.5e-28)
		tmp = Float64(Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m) * Float64(Float64(l * sqrt(2.0)) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * sin(k_m)) ^ 2.0) / Float64(cos(k_m) * Float64((l ^ 2.0) / t_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 <= 1.5e-28)
		tmp = ((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0)) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / (((k_m * sin(k_m)) ^ 2.0) / (cos(k_m) * ((l ^ 2.0) / t_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, 1.5e-28], N[Power[N[(N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000001e-28

   1. Initial program 32.1%

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

      1. associate-/r* 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-*l* 32.1%

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

      4. associate-*l/ 32.2%

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

      5. +-commutative 32.2%

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

      6. unpow2 32.2%

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

      7. sqr-neg 32.2%

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

      8. distribute-frac-neg 32.2%

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

      9. distribute-frac-neg 32.2%

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

      10. unpow2 32.2%

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

      11. associate--l+ 39.7%

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

      12. metadata-eval 39.7%

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

      13. +-rgt-identity 39.7%

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

      14. unpow2 39.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.7%

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

      16. distribute-frac-neg 39.7%

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

   3. Simplified 39.7%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 20.9%

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

      1. unpow2 20.9%

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

      2. associate-/l/ 20.9%

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

      3. *-commutative 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in k around inf 41.3%

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

      1. associate-*l/ 40.7%

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

      2. *-commutative 40.7%

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

      3. times-frac 43.3%

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

   10. Simplified 43.3%

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

   11. Taylor expanded in k around 0 35.8%

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

   if 1.50000000000000001e-28 < k

   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. associate-/r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*l* 29.0%

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

      4. associate-*l/ 29.0%

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

      5. +-commutative 29.0%

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

      6. unpow2 29.0%

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

      7. sqr-neg 29.0%

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

      8. distribute-frac-neg 29.0%

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

      9. distribute-frac-neg 29.0%

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

      10. unpow2 29.0%

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

      11. associate--l+ 46.0%

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

      12. metadata-eval 46.0%

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

      13. +-rgt-identity 46.0%

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

      14. unpow2 46.0%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 46.0%

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

      16. distribute-frac-neg 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in k around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 78.8%

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

      3. associate-/r* 78.8%

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

   6. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 71.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 59.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}\right)} - 1} \]

      3. frac-times 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}}\right)} - 1 \]

      4. associate-/l* 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{t}{\cos k}}}}{{k}^{2} \cdot {\sin k}^{2}}\right)} - 1 \]

      5. pow-prod-down 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{\color{blue}{{\left(k \cdot \sin k\right)}^{2}}}\right)} - 1 \]

   8. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{{\left(k \cdot \sin k\right)}^{2}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 71.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{{\left(k \cdot \sin k\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 78.8%

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

      3. associate-/l* 78.9%

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

      4. associate-/r/ 78.8%

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

   10. Simplified 78.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k}{t}}}{k} \cdot \frac{\ell \cdot \sqrt{2}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k \cdot \frac{{\ell}^{2}}{t}}}\\ \end{array} \]

Alternative 7: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.9 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k_m}{t_m}}}{k_m} \cdot \frac{\ell \cdot \sqrt{2}}{k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t_m}{\cos k_m}}}{{\left(k_m \cdot \sin k_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.9e-28)
    (pow (* (/ (sqrt (/ (cos k_m) t_m)) k_m) (/ (* l (sqrt 2.0)) k_m)) 2.0)
    (/
     (* 2.0 (/ (pow l 2.0) (/ t_m (cos k_m))))
     (pow (* k_m (sin k_m)) 2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-28) {
		tmp = pow(((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0)) / k_m)), 2.0);
	} else {
		tmp = (2.0 * (pow(l, 2.0) / (t_m / cos(k_m)))) / pow((k_m * sin(k_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-28) then
        tmp = ((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0d0)) / k_m)) ** 2.0d0
    else
        tmp = (2.0d0 * ((l ** 2.0d0) / (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 (k_m <= 1.9e-28) {
		tmp = Math.pow(((Math.sqrt((Math.cos(k_m) / t_m)) / k_m) * ((l * Math.sqrt(2.0)) / k_m)), 2.0);
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) / (t_m / Math.cos(k_m)))) / Math.pow((k_m * Math.sin(k_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.9e-28:
		tmp = math.pow(((math.sqrt((math.cos(k_m) / t_m)) / k_m) * ((l * math.sqrt(2.0)) / k_m)), 2.0)
	else:
		tmp = (2.0 * (math.pow(l, 2.0) / (t_m / math.cos(k_m)))) / math.pow((k_m * math.sin(k_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-28)
		tmp = Float64(Float64(sqrt(Float64(cos(k_m) / t_m)) / k_m) * Float64(Float64(l * sqrt(2.0)) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) / 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 (k_m <= 1.9e-28)
		tmp = ((sqrt((cos(k_m) / t_m)) / k_m) * ((l * sqrt(2.0)) / k_m)) ^ 2.0;
	else
		tmp = (2.0 * ((l ^ 2.0) / (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[k$95$m, 1.9e-28], N[Power[N[(N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.90000000000000005e-28

   1. Initial program 32.1%

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

      1. associate-/r* 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-*l* 32.1%

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

      4. associate-*l/ 32.2%

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

      5. +-commutative 32.2%

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

      6. unpow2 32.2%

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

      7. sqr-neg 32.2%

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

      8. distribute-frac-neg 32.2%

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

      9. distribute-frac-neg 32.2%

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

      10. unpow2 32.2%

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

      11. associate--l+ 39.7%

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

      12. metadata-eval 39.7%

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

      13. +-rgt-identity 39.7%

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

      14. unpow2 39.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.7%

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

      16. distribute-frac-neg 39.7%

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

   3. Simplified 39.7%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 20.9%

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

      1. unpow2 20.9%

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

      2. associate-/l/ 20.9%

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

      3. *-commutative 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in k around inf 41.3%

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

      1. associate-*l/ 40.7%

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

      2. *-commutative 40.7%

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

      3. times-frac 43.3%

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

   10. Simplified 43.3%

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

   11. Taylor expanded in k around 0 35.8%

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

   if 1.90000000000000005e-28 < k

   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. associate-/r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*l* 29.0%

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

      4. associate-*l/ 29.0%

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

      5. +-commutative 29.0%

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

      6. unpow2 29.0%

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

      7. sqr-neg 29.0%

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

      8. distribute-frac-neg 29.0%

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

      9. distribute-frac-neg 29.0%

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

      10. unpow2 29.0%

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

      11. associate--l+ 46.0%

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

      12. metadata-eval 46.0%

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

      13. +-rgt-identity 46.0%

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

      14. unpow2 46.0%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 46.0%

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

      16. distribute-frac-neg 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in k around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 78.8%

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

      3. associate-/r* 78.8%

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

   6. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 71.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 59.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}\right)} - 1} \]

      3. frac-times 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}}\right)} - 1 \]

      4. associate-/l* 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{t}{\cos k}}}}{{k}^{2} \cdot {\sin k}^{2}}\right)} - 1 \]

      5. pow-prod-down 59.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{\color{blue}{{\left(k \cdot \sin k\right)}^{2}}}\right)} - 1 \]

   8. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{{\left(k \cdot \sin k\right)}^{2}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 71.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{{\left(k \cdot \sin k\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 78.8%

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

   10. Simplified 78.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\cos k}{t}}}{k} \cdot \frac{\ell \cdot \sqrt{2}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2}}{\frac{t}{\cos k}}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]

Alternative 8: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 21000000:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k_m}^{2}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;k_m \leq 1.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\right) \cdot \left(\frac{{t_m}^{3}}{\ell} \cdot \frac{\sin k_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 21000000.0)
    (pow (* (/ (* l (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= k_m 1.3e+53)
      (/
       2.0
       (*
        (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))
        (* (/ (pow t_m 3.0) l) (/ (sin k_m) l))))
      (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 21000000.0) {
		tmp = pow((((l * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if (k_m <= 1.3e+53) {
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((pow(t_m, 3.0) / l) * (sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_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 <= 21000000.0d0) then
        tmp = (((l * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if (k_m <= 1.3d+53) then
        tmp = 2.0d0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ** 3.0d0) / l) * (sin(k_m) / l)))
    else
        tmp = (-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_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 <= 21000000.0) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (k_m <= 1.3e+53) {
		tmp = 2.0 / ((Math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((Math.pow(t_m, 3.0) / l) * (Math.sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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 <= 21000000.0:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	elif k_m <= 1.3e+53:
		tmp = 2.0 / ((math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((math.pow(t_m, 3.0) / l) * (math.sin(k_m) / l)))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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)
	tmp = 0.0
	if (k_m <= 21000000.0)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (k_m <= 1.3e+53)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m)))) * Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k_m) / l))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_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 <= 21000000.0)
		tmp = (((l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif (k_m <= 1.3e+53)
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ^ 3.0) / l) * (sin(k_m) / l)));
	else
		tmp = -0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_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, 21000000.0], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.3e+53], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.1e7

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around 0 35.0%

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

   if 2.1e7 < k < 1.29999999999999999e53

   1. Initial program 13.9%

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

      1. associate-*l* 13.9%

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

      2. associate-*l/ 13.9%

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

      3. associate--l+ 13.9%

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

   3. Simplified 13.9%

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

      1. associate-+r- 13.9%

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

      2. add-exp-log 12.8%

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

      3. log1p-udef 12.8%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 43.2%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 44.2%

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

      6. unpow2 44.2%

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

      7. clear-num 44.1%

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

      8. frac-times 44.4%

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

      9. *-un-lft-identity 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. times-frac 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 1.29999999999999999e53 < k

   1. Initial program 34.3%

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

      1. associate-/r* 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*l* 34.3%

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

      4. associate-*l/ 34.3%

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

      5. +-commutative 34.3%

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

      6. unpow2 34.3%

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

      7. sqr-neg 34.3%

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

      8. distribute-frac-neg 34.3%

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

      9. distribute-frac-neg 34.3%

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

      10. unpow2 34.3%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 59.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 61.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 61.9%

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

      3. metadata-eval 61.9%

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

   9. Applied egg-rr 61.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 60.9%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 60.9%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 60.9%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 61.9%

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

       3. associate-*r/ 61.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 61.8%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 63.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 21000000:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 9: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 48000000:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \left(\ell \cdot \left(\sqrt{2} \cdot {k_m}^{-2}\right)\right)\right)}^{2}\\ \mathbf{elif}\;k_m \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\right) \cdot \left(\frac{{t_m}^{3}}{\ell} \cdot \frac{\sin k_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 48000000.0)
    (pow (* (pow t_m -0.5) (* l (* (sqrt 2.0) (pow k_m -2.0)))) 2.0)
    (if (<= k_m 6.8e+52)
      (/
       2.0
       (*
        (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))
        (* (/ (pow t_m 3.0) l) (/ (sin k_m) l))))
      (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 48000000.0) {
		tmp = pow((pow(t_m, -0.5) * (l * (sqrt(2.0) * pow(k_m, -2.0)))), 2.0);
	} else if (k_m <= 6.8e+52) {
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((pow(t_m, 3.0) / l) * (sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_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 <= 48000000.0d0) then
        tmp = ((t_m ** (-0.5d0)) * (l * (sqrt(2.0d0) * (k_m ** (-2.0d0))))) ** 2.0d0
    else if (k_m <= 6.8d+52) then
        tmp = 2.0d0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ** 3.0d0) / l) * (sin(k_m) / l)))
    else
        tmp = (-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_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 <= 48000000.0) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l * (Math.sqrt(2.0) * Math.pow(k_m, -2.0)))), 2.0);
	} else if (k_m <= 6.8e+52) {
		tmp = 2.0 / ((Math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((Math.pow(t_m, 3.0) / l) * (Math.sin(k_m) / l)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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 <= 48000000.0:
		tmp = math.pow((math.pow(t_m, -0.5) * (l * (math.sqrt(2.0) * math.pow(k_m, -2.0)))), 2.0)
	elif k_m <= 6.8e+52:
		tmp = 2.0 / ((math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((math.pow(t_m, 3.0) / l) * (math.sin(k_m) / l)))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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)
	tmp = 0.0
	if (k_m <= 48000000.0)
		tmp = Float64((t_m ^ -0.5) * Float64(l * Float64(sqrt(2.0) * (k_m ^ -2.0)))) ^ 2.0;
	elseif (k_m <= 6.8e+52)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m)))) * Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k_m) / l))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_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 <= 48000000.0)
		tmp = ((t_m ^ -0.5) * (l * (sqrt(2.0) * (k_m ^ -2.0)))) ^ 2.0;
	elseif (k_m <= 6.8e+52)
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ^ 3.0) / l) * (sin(k_m) / l)));
	else
		tmp = -0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_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, 48000000.0], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 6.8e+52], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.8e7

   1. Initial program 32.0%

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

      1. associate-/r* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

      4. associate-*l/ 32.0%

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

      5. +-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. sqr-neg 32.0%

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

      8. distribute-frac-neg 32.0%

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

      9. distribute-frac-neg 32.0%

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

      10. unpow2 32.0%

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

      11. associate--l+ 39.4%

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

      12. metadata-eval 39.4%

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

      13. +-rgt-identity 39.4%

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

      14. unpow2 39.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.4%

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

      16. distribute-frac-neg 39.4%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 25.2%

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

   5. Applied egg-rr 21.5%

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

      1. unpow2 21.5%

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

      2. associate-/l/ 21.5%

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

      3. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around 0 35.0%

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

      1. add-log-exp 28.8%

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

      2. exp-prod 38.6%

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

      3. div-inv 38.6%

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

      4. pow-flip 38.6%

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

      5. metadata-eval 38.6%

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

      6. exp-prod 34.1%

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

      7. exp-prod 34.1%

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

      8. pow1/2 34.1%

         \[\leadsto {\log \left({\left({\left({\left(e^{\ell}\right)}^{\left(\sqrt{2}\right)}\right)}^{\left({k}^{-2}\right)}\right)}^{\color{blue}{\left({\left(\frac{1}{t}\right)}^{0.5}\right)}}\right)}^{2} \]

      9. inv-pow 34.1%

         \[\leadsto {\log \left({\left({\left({\left(e^{\ell}\right)}^{\left(\sqrt{2}\right)}\right)}^{\left({k}^{-2}\right)}\right)}^{\left({\color{blue}{\left({t}^{-1}\right)}}^{0.5}\right)}\right)}^{2} \]

      10. pow-pow 34.1%

          \[\leadsto {\log \left({\left({\left({\left(e^{\ell}\right)}^{\left(\sqrt{2}\right)}\right)}^{\left({k}^{-2}\right)}\right)}^{\color{blue}{\left({t}^{\left(-1 \cdot 0.5\right)}\right)}}\right)}^{2} \]

      11. metadata-eval 34.1%

          \[\leadsto {\log \left({\left({\left({\left(e^{\ell}\right)}^{\left(\sqrt{2}\right)}\right)}^{\left({k}^{-2}\right)}\right)}^{\left({t}^{\color{blue}{-0.5}}\right)}\right)}^{2} \]

   10. Applied egg-rr 34.1%

       \[\leadsto {\color{blue}{\log \left({\left({\left({\left(e^{\ell}\right)}^{\left(\sqrt{2}\right)}\right)}^{\left({k}^{-2}\right)}\right)}^{\left({t}^{-0.5}\right)}\right)}}^{2} \]
   11. Step-by-step derivation

       1. log-pow 23.4%

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

       2. log-pow 21.3%

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

       3. log-pow 21.3%

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

       4. rem-log-exp 35.0%

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

       5. *-commutative 35.0%

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

       6. *-commutative 35.0%

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

       7. associate-*l* 35.0%

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

   12. Simplified 35.0%

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

   if 4.8e7 < k < 6.8e52

   1. Initial program 13.9%

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

      1. associate-*l* 13.9%

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

      2. associate-*l/ 13.9%

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

      3. associate--l+ 13.9%

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

   3. Simplified 13.9%

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

      1. associate-+r- 13.9%

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

      2. add-exp-log 12.8%

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

      3. log1p-udef 12.8%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 43.2%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 44.2%

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

      6. unpow2 44.2%

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

      7. clear-num 44.1%

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

      8. frac-times 44.4%

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

      9. *-un-lft-identity 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. times-frac 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 6.8e52 < k

   1. Initial program 34.3%

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

      1. associate-/r* 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*l* 34.3%

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

      4. associate-*l/ 34.3%

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

      5. +-commutative 34.3%

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

      6. unpow2 34.3%

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

      7. sqr-neg 34.3%

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

      8. distribute-frac-neg 34.3%

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

      9. distribute-frac-neg 34.3%

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

      10. unpow2 34.3%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 59.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 61.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 61.8%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 61.9%

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

      3. metadata-eval 61.9%

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

   9. Applied egg-rr 61.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 60.9%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 60.9%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 60.9%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 61.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 61.9%

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

       3. associate-*r/ 61.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 61.8%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 63.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 48000000:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)\right)}^{2}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 10: 72.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{t_m} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k_m}^{-2}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 1.28 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\right) \cdot \left(\frac{{t_m}^{3}}{\ell} \cdot \frac{\sin k_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k_m}^{2}}{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e-93)
    (* (/ 1.0 t_m) (pow (* (* l (sqrt 2.0)) (pow k_m -2.0)) 2.0))
    (if (<= t_m 1.28e+95)
      (/
       2.0
       (*
        (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))
        (* (/ (pow t_m 3.0) l) (/ (sin k_m) l))))
      (/ 2.0 (/ (* t_m (pow k_m 2.0)) (pow (/ l (fabs 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 (t_m <= 5.4e-93) {
		tmp = (1.0 / t_m) * pow(((l * sqrt(2.0)) * pow(k_m, -2.0)), 2.0);
	} else if (t_m <= 1.28e+95) {
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((pow(t_m, 3.0) / l) * (sin(k_m) / l)));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 2.0)) / pow((l / fabs(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 (t_m <= 5.4d-93) then
        tmp = (1.0d0 / t_m) * (((l * sqrt(2.0d0)) * (k_m ** (-2.0d0))) ** 2.0d0)
    else if (t_m <= 1.28d+95) then
        tmp = 2.0d0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ** 3.0d0) / l) * (sin(k_m) / l)))
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 2.0d0)) / ((l / abs(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 (t_m <= 5.4e-93) {
		tmp = (1.0 / t_m) * Math.pow(((l * Math.sqrt(2.0)) * Math.pow(k_m, -2.0)), 2.0);
	} else if (t_m <= 1.28e+95) {
		tmp = 2.0 / ((Math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((Math.pow(t_m, 3.0) / l) * (Math.sin(k_m) / l)));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 2.0)) / Math.pow((l / Math.abs(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 t_m <= 5.4e-93:
		tmp = (1.0 / t_m) * math.pow(((l * math.sqrt(2.0)) * math.pow(k_m, -2.0)), 2.0)
	elif t_m <= 1.28e+95:
		tmp = 2.0 / ((math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * ((math.pow(t_m, 3.0) / l) * (math.sin(k_m) / l)))
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 2.0)) / math.pow((l / math.fabs(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 (t_m <= 5.4e-93)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(Float64(l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0));
	elseif (t_m <= 1.28e+95)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m)))) * Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 2.0)) / (Float64(l / abs(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 (t_m <= 5.4e-93)
		tmp = (1.0 / t_m) * (((l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0);
	elseif (t_m <= 1.28e+95)
		tmp = 2.0 / ((tan(k_m) * (k_m / (t_m * (t_m / k_m)))) * (((t_m ^ 3.0) / l) * (sin(k_m) / l)));
	else
		tmp = 2.0 / ((t_m * (k_m ^ 2.0)) / ((l / abs(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[t$95$m, 5.4e-93], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.28e+95], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.4000000000000002e-93

   1. Initial program 30.7%

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

      1. associate-/r* 30.7%

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

      2. *-commutative 30.7%

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

      3. associate-*l* 30.7%

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

      4. associate-*l/ 30.7%

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

      5. +-commutative 30.7%

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

      6. unpow2 30.7%

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

      7. sqr-neg 30.7%

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

      8. distribute-frac-neg 30.7%

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

      9. distribute-frac-neg 30.7%

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

      10. unpow2 30.7%

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

      11. associate--l+ 39.0%

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

      12. metadata-eval 39.0%

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

      13. +-rgt-identity 39.0%

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

      14. unpow2 39.0%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 39.0%

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

      16. distribute-frac-neg 39.0%

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

   3. Simplified 39.0%

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

      1. add-sqr-sqrt 23.3%

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

   5. Applied egg-rr 7.9%

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

      1. unpow2 7.9%

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

      2. associate-/l/ 7.9%

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

      3. *-commutative 7.9%

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

   7. Simplified 7.9%

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

   8. Taylor expanded in k around 0 12.6%

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

      1. unpow-prod-down 12.6%

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

      2. div-inv 12.6%

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

      3. pow-flip 12.6%

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

      4. metadata-eval 12.6%

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

      5. pow2 12.6%

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

      6. add-sqr-sqrt 70.1%

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

   10. Applied egg-rr 70.1%

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

   if 5.4000000000000002e-93 < t < 1.28000000000000006e95

   1. Initial program 52.6%

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

      1. associate-*l* 52.7%

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

      2. associate-*l/ 52.7%

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

      3. associate--l+ 52.7%

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

   3. Simplified 52.7%

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

      1. associate-+r- 52.7%

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

      2. add-exp-log 52.3%

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

      3. log1p-udef 52.3%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 65.6%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 66.0%

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

      6. unpow2 66.0%

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

      7. clear-num 66.0%

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

      8. frac-times 66.1%

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

      9. *-un-lft-identity 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. times-frac 71.7%

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

   7. Applied egg-rr 71.7%

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

   if 1.28000000000000006e95 < t

   1. Initial program 14.6%

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

      1. associate-*l* 14.6%

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

      2. associate-*l/ 14.6%

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

      3. associate--l+ 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in t around 0 73.8%

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

      1. times-frac 73.9%

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

   6. Simplified 73.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. clear-num 74.0%

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

      2. inv-pow 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. unpow-1 74.0%

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

   10. Simplified 74.0%

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

       1. inv-pow 74.0%

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

       2. add-sqr-sqrt 73.8%

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

       3. unpow-prod-down 73.8%

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

       4. div-inv 73.8%

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

       5. sqrt-prod 73.9%

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

       6. unpow2 73.9%

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

       7. sqrt-prod 34.9%

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

       8. add-sqr-sqrt 49.8%

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

       9. pow-flip 49.8%

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

       10. metadata-eval 49.8%

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

   12. Applied egg-rr 84.2%

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

       1. pow-sqr 84.1%

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

       2. sqr-pow 84.1%

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

       3. rem-sqrt-square 87.9%

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

       4. metadata-eval 87.9%

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

       5. unpow-1 87.9%

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

       6. metadata-eval 87.9%

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

   14. Simplified 87.9%

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

   15. Taylor expanded in k around 0 67.0%

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

       1. *-commutative 67.0%

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

       2. unpow2 67.0%

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

       3. fabs-div 67.0%

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

       4. metadata-eval 67.0%

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

       5. unpow2 67.0%

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

       6. swap-sqr 76.6%

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

       7. unpow1 76.6%

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

       8. pow-plus 76.6%

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

       9. associate-*r/ 76.6%

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

       10. *-rgt-identity 76.6%

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

       11. metadata-eval 76.6%

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

   17. Simplified 76.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k}^{-2}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}}\\ \end{array} \]

Alternative 11: 69.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{t_m} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k_m}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{2}{t_m}}{{k_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-49)
    (* (/ 1.0 t_m) (pow (* (* l (sqrt 2.0)) (pow k_m -2.0)) 2.0))
    (/ (* (pow l 2.0) (/ 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) {
	double tmp;
	if ((l * l) <= 2e-49) {
		tmp = (1.0 / t_m) * pow(((l * sqrt(2.0)) * pow(k_m, -2.0)), 2.0);
	} else {
		tmp = (pow(l, 2.0) * (2.0 / t_m)) / pow(k_m, 4.0);
	}
	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) <= 2d-49) then
        tmp = (1.0d0 / t_m) * (((l * sqrt(2.0d0)) * (k_m ** (-2.0d0))) ** 2.0d0)
    else
        tmp = ((l ** 2.0d0) * (2.0d0 / t_m)) / (k_m ** 4.0d0)
    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) <= 2e-49) {
		tmp = (1.0 / t_m) * Math.pow(((l * Math.sqrt(2.0)) * Math.pow(k_m, -2.0)), 2.0);
	} else {
		tmp = (Math.pow(l, 2.0) * (2.0 / t_m)) / Math.pow(k_m, 4.0);
	}
	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) <= 2e-49:
		tmp = (1.0 / t_m) * math.pow(((l * math.sqrt(2.0)) * math.pow(k_m, -2.0)), 2.0)
	else:
		tmp = (math.pow(l, 2.0) * (2.0 / t_m)) / math.pow(k_m, 4.0)
	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) <= 2e-49)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(Float64(l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0));
	else
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 / t_m)) / (k_m ^ 4.0));
	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) <= 2e-49)
		tmp = (1.0 / t_m) * (((l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0);
	else
		tmp = ((l ^ 2.0) * (2.0 / t_m)) / (k_m ^ 4.0);
	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], 2e-49], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999987e-49

   1. Initial program 28.0%

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

      1. associate-/r* 28.0%

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

      2. *-commutative 28.0%

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

      3. associate-*l* 28.0%

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

      4. associate-*l/ 28.0%

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

      5. +-commutative 28.0%

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

      6. unpow2 28.0%

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

      7. sqr-neg 28.0%

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

      8. distribute-frac-neg 28.0%

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

      9. distribute-frac-neg 28.0%

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

      10. unpow2 28.0%

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

      11. associate--l+ 41.0%

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

      12. metadata-eval 41.0%

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

      13. +-rgt-identity 41.0%

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

      14. unpow2 41.0%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 41.0%

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

      16. distribute-frac-neg 41.0%

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

   3. Simplified 41.0%

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

      1. add-sqr-sqrt 37.0%

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

   5. Applied egg-rr 13.8%

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

      1. unpow2 13.8%

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

      2. associate-/l/ 13.8%

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

      3. *-commutative 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in k around 0 40.0%

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

      1. unpow-prod-down 39.3%

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

      2. div-inv 39.4%

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

      3. pow-flip 39.4%

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

      4. metadata-eval 39.4%

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

      5. pow2 39.4%

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

      6. add-sqr-sqrt 89.7%

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

   10. Applied egg-rr 89.7%

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

   if 1.99999999999999987e-49 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l/ 34.4%

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

      3. associate--l+ 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in k around 0 54.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 54.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{{k}^{4} \cdot t} \cdot \sqrt[3]{{k}^{4} \cdot t}\right) \cdot \sqrt[3]{{k}^{4} \cdot t}}}{{\ell}^{2}}} \]

      2. pow3 54.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{{k}^{4} \cdot t}\right)}^{3}}}{{\ell}^{2}}} \]

      3. *-commutative 54.4%

         \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}}{{\ell}^{2}}} \]

   6. Applied egg-rr 54.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}}{{\ell}^{2}}} \]

   7. Taylor expanded in t around 0 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l/ 54.5%

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

      4. associate-/r* 54.5%

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

      5. associate-*l/ 55.8%

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

   9. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{t} \cdot {\ell}^{2}}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{2}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 12: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{t_m} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k_m}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k_m}^{2}}{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+76)
    (* (/ 1.0 t_m) (pow (* (* l (sqrt 2.0)) (pow k_m -2.0)) 2.0))
    (/ 2.0 (/ (* t_m (pow k_m 2.0)) (pow (/ l (fabs 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 (t_m <= 2.1e+76) {
		tmp = (1.0 / t_m) * pow(((l * sqrt(2.0)) * pow(k_m, -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 2.0)) / pow((l / fabs(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 (t_m <= 2.1d+76) then
        tmp = (1.0d0 / t_m) * (((l * sqrt(2.0d0)) * (k_m ** (-2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 2.0d0)) / ((l / abs(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 (t_m <= 2.1e+76) {
		tmp = (1.0 / t_m) * Math.pow(((l * Math.sqrt(2.0)) * Math.pow(k_m, -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 2.0)) / Math.pow((l / Math.abs(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 t_m <= 2.1e+76:
		tmp = (1.0 / t_m) * math.pow(((l * math.sqrt(2.0)) * math.pow(k_m, -2.0)), 2.0)
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 2.0)) / math.pow((l / math.fabs(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 (t_m <= 2.1e+76)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(Float64(l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 2.0)) / (Float64(l / abs(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 (t_m <= 2.1e+76)
		tmp = (1.0 / t_m) * (((l * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0);
	else
		tmp = 2.0 / ((t_m * (k_m ^ 2.0)) / ((l / abs(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[t$95$m, 2.1e+76], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.10000000000000007e76

   1. Initial program 33.8%

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

      1. associate-/r* 33.8%

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

      2. *-commutative 33.8%

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

      3. associate-*l* 33.8%

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

      4. associate-*l/ 33.9%

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

      5. +-commutative 33.9%

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

      6. unpow2 33.9%

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

      7. sqr-neg 33.9%

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

      8. distribute-frac-neg 33.9%

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

      9. distribute-frac-neg 33.9%

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

      10. unpow2 33.9%

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

      11. associate--l+ 42.3%

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

      12. metadata-eval 42.3%

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

      13. +-rgt-identity 42.3%

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

      14. unpow2 42.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 42.3%

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

      16. distribute-frac-neg 42.3%

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

   3. Simplified 42.3%

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

      1. add-sqr-sqrt 29.0%

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

   5. Applied egg-rr 14.7%

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

      1. unpow2 14.7%

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

      2. associate-/l/ 14.7%

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

      3. *-commutative 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in k around 0 21.4%

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

      1. unpow-prod-down 21.0%

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

      2. div-inv 21.0%

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

      3. pow-flip 21.0%

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

      4. metadata-eval 21.0%

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

      5. pow2 21.0%

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

      6. add-sqr-sqrt 69.7%

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

   10. Applied egg-rr 69.7%

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

   if 2.10000000000000007e76 < t

   1. Initial program 19.7%

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

      1. associate-*l* 19.7%

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

      2. associate-*l/ 19.7%

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

      3. associate--l+ 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in t around 0 76.5%

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

      1. times-frac 76.7%

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

   6. Simplified 76.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. clear-num 76.7%

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

      2. inv-pow 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. unpow-1 76.7%

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

   10. Simplified 76.7%

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

       1. inv-pow 76.7%

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

       2. add-sqr-sqrt 76.6%

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

       3. unpow-prod-down 76.5%

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

       4. div-inv 76.6%

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

       5. sqrt-prod 76.6%

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

       6. unpow2 76.6%

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

       7. sqrt-prod 35.4%

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

       8. add-sqr-sqrt 50.9%

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

       9. pow-flip 50.9%

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

       10. metadata-eval 50.9%

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

   12. Applied egg-rr 85.9%

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

       1. pow-sqr 85.8%

          \[\leadsto \frac{2}{\color{blue}{{\left(\ell \cdot \sqrt{{k}^{-2}}\right)}^{\left(2 \cdot -1\right)}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       2. sqr-pow 85.8%

          \[\leadsto \frac{2}{{\left(\ell \cdot \sqrt{\color{blue}{{k}^{\left(\frac{-2}{2}\right)} \cdot {k}^{\left(\frac{-2}{2}\right)}}}\right)}^{\left(2 \cdot -1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       3. rem-sqrt-square 89.1%

          \[\leadsto \frac{2}{{\left(\ell \cdot \color{blue}{\left|{k}^{\left(\frac{-2}{2}\right)}\right|}\right)}^{\left(2 \cdot -1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       4. metadata-eval 89.1%

          \[\leadsto \frac{2}{{\left(\ell \cdot \left|{k}^{\color{blue}{-1}}\right|\right)}^{\left(2 \cdot -1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       5. unpow-1 89.1%

          \[\leadsto \frac{2}{{\left(\ell \cdot \left|\color{blue}{\frac{1}{k}}\right|\right)}^{\left(2 \cdot -1\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       6. metadata-eval 89.1%

          \[\leadsto \frac{2}{{\left(\ell \cdot \left|\frac{1}{k}\right|\right)}^{\color{blue}{-2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   14. Simplified 89.1%

       \[\leadsto \frac{2}{\color{blue}{{\left(\ell \cdot \left|\frac{1}{k}\right|\right)}^{-2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   15. Taylor expanded in k around 0 66.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2} \cdot {\left(\left|\frac{1}{k}\right|\right)}^{2}}}} \]
   16. Step-by-step derivation

       1. *-commutative 66.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2} \cdot {\left(\left|\frac{1}{k}\right|\right)}^{2}}} \]

       2. unpow2 66.5%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot {\left(\left|\frac{1}{k}\right|\right)}^{2}}} \]

       3. fabs-div 66.5%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\left(\ell \cdot \ell\right) \cdot {\color{blue}{\left(\frac{\left|1\right|}{\left|k\right|}\right)}}^{2}}} \]

       4. metadata-eval 66.5%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\left(\ell \cdot \ell\right) \cdot {\left(\frac{\color{blue}{1}}{\left|k\right|}\right)}^{2}}} \]

       5. unpow2 66.5%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{\left|k\right|} \cdot \frac{1}{\left|k\right|}\right)}}} \]

       6. swap-sqr 75.0%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\left(\ell \cdot \frac{1}{\left|k\right|}\right) \cdot \left(\ell \cdot \frac{1}{\left|k\right|}\right)}}} \]

       7. unpow1 75.0%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{{\left(\ell \cdot \frac{1}{\left|k\right|}\right)}^{1}} \cdot \left(\ell \cdot \frac{1}{\left|k\right|}\right)}} \]

       8. pow-plus 75.0%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{{\left(\ell \cdot \frac{1}{\left|k\right|}\right)}^{\left(1 + 1\right)}}}} \]

       9. associate-*r/ 75.0%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{{\color{blue}{\left(\frac{\ell \cdot 1}{\left|k\right|}\right)}}^{\left(1 + 1\right)}}} \]

       10. *-rgt-identity 75.0%

           \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{{\left(\frac{\color{blue}{\ell}}{\left|k\right|}\right)}^{\left(1 + 1\right)}}} \]

       11. metadata-eval 75.0%

           \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{{\left(\frac{\ell}{\left|k\right|}\right)}^{\color{blue}{2}}}} \]

   17. Simplified 75.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{2}}{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}}\\ \end{array} \]

Alternative 13: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t_m \cdot {k_m}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.4e+14)
    (pow (* l (sqrt (/ 2.0 (* t_m (pow k_m 4.0))))) 2.0)
    (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e+14) {
		tmp = pow((l * sqrt((2.0 / (t_m * pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_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 <= 5.4d+14) then
        tmp = (l * sqrt((2.0d0 / (t_m * (k_m ** 4.0d0))))) ** 2.0d0
    else
        tmp = (-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_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 <= 5.4e+14) {
		tmp = Math.pow((l * Math.sqrt((2.0 / (t_m * Math.pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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 <= 5.4e+14:
		tmp = math.pow((l * math.sqrt((2.0 / (t_m * math.pow(k_m, 4.0))))), 2.0)
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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)
	tmp = 0.0
	if (k_m <= 5.4e+14)
		tmp = Float64(l * sqrt(Float64(2.0 / Float64(t_m * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_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 <= 5.4e+14)
		tmp = (l * sqrt((2.0 / (t_m * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = -0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_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, 5.4e+14], N[Power[N[(l * N[Sqrt[N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.4e14

   1. Initial program 31.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 31.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-*l/ 31.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate--l+ 31.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 31.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 59.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 59.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{{k}^{4} \cdot t} \cdot \sqrt[3]{{k}^{4} \cdot t}\right) \cdot \sqrt[3]{{k}^{4} \cdot t}}}{{\ell}^{2}}} \]

      2. pow3 59.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{{k}^{4} \cdot t}\right)}^{3}}}{{\ell}^{2}}} \]

      3. *-commutative 59.3%

         \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}}{{\ell}^{2}}} \]

   6. Applied egg-rr 59.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 38.2%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}{{\ell}^{2}}}} \cdot \sqrt{\frac{2}{\frac{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}{{\ell}^{2}}}}} \]

      2. pow2 38.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{\frac{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}{{\ell}^{2}}}}\right)}^{2}} \]

      3. rem-cube-cbrt 38.2%

         \[\leadsto {\left(\sqrt{\frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}}}\right)}^{2} \]

      4. associate-/r/ 38.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}}}\right)}^{2} \]

      5. sqrt-prod 36.6%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{{\ell}^{2}}\right)}}^{2} \]

      6. unpow2 36.6%

         \[\leadsto {\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{\color{blue}{\ell \cdot \ell}}\right)}^{2} \]

      7. sqrt-prod 17.1%

         \[\leadsto {\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right)}^{2} \]

      8. add-sqr-sqrt 40.9%

         \[\leadsto {\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \color{blue}{\ell}\right)}^{2} \]

   8. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \ell\right)}^{2}} \]

   if 5.4e14 < k

   1. Initial program 30.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 30.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. *-commutative 30.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l* 30.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-*l/ 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \color{blue}{\frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 30.1%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 30.1%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

      16. distribute-frac-neg 45.7%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(-\frac{k}{t}\right) \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 50.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 55.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 55.7%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 55.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 55.7%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 55.8%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \]

      3. metadata-eval 55.8%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-2}}\right) \]

   9. Applied egg-rr 55.8%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 51.5%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 51.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 51.1%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 51.1%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 51.5%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 55.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

       3. associate-*r/ 55.6%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 55.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 56.8%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 14: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t_m \cdot {k_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.25e+48)
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k_m 4.0))))
    (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+48) {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_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 <= 2.25d+48) then
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_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 <= 2.25e+48) {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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 <= 2.25e+48:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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)
	tmp = 0.0
	if (k_m <= 2.25e+48)
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_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 <= 2.25e+48)
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = -0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_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, 2.25e+48], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.24999999999999998e48

   1. Initial program 30.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-*l/ 30.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate--l+ 30.3%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 57.3%

         \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t} \cdot {\ell}^{2}} \]

      2. *-commutative 57.3%

         \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot {\ell}^{2} \]

   6. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}} \]

   if 2.24999999999999998e48 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 35.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. *-commutative 35.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l* 35.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \color{blue}{\frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 35.5%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

      16. distribute-frac-neg 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(-\frac{k}{t}\right) \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 58.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 61.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 61.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 61.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 61.1%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-2}}\right) \]

   9. Applied egg-rr 61.1%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 59.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 59.7%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 59.7%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 59.7%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 59.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 61.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

       3. associate-*r/ 60.9%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 60.9%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 62.5%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 15: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k_m}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.25e+48)
    (/ 2.0 (/ (* t_m (pow k_m 4.0)) (pow l 2.0)))
    (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+48) {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / pow(l, 2.0));
	} else {
		tmp = -0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_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 <= 2.25d+48) then
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / (l ** 2.0d0))
    else
        tmp = (-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_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 <= 2.25e+48) {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_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 <= 2.25e+48:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / 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)
	tmp = 0.0
	if (k_m <= 2.25e+48)
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / (l ^ 2.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_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 <= 2.25e+48)
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / (l ^ 2.0));
	else
		tmp = -0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_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, 2.25e+48], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.24999999999999998e48

   1. Initial program 30.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-*l/ 30.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate--l+ 30.3%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   if 2.24999999999999998e48 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 35.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. *-commutative 35.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l* 35.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \color{blue}{\frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 35.5%

         \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 35.5%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

      16. distribute-frac-neg 49.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(-\frac{k}{t}\right) \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 58.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 61.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      2. associate-/r* 61.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. div-inv 61.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

      2. pow-flip 61.1%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-2}}\right) \]

   9. Applied egg-rr 61.1%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 59.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

       2. expm1-udef 59.7%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

       3. *-commutative 59.7%

          \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

   11. Applied egg-rr 59.7%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 59.8%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

       2. expm1-log1p 61.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

       3. associate-*r/ 60.9%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

       4. *-commutative 60.9%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

   13. Simplified 62.5%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}\\ \end{array} \]

Alternative 16: 34.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k_m\right|}\right)}^{2}}{t_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* -0.3333333333333333 (/ (pow (/ l (fabs k_m)) 2.0) t_m))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (-0.3333333333333333 * (pow((l / fabs(k_m)), 2.0) / t_m));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((-0.3333333333333333d0) * (((l / abs(k_m)) ** 2.0d0) / t_m))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (-0.3333333333333333 * (Math.pow((l / Math.abs(k_m)), 2.0) / t_m));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (-0.3333333333333333 * (math.pow((l / math.fabs(k_m)), 2.0) / t_m))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(-0.3333333333333333 * Float64((Float64(l / abs(k_m)) ^ 2.0) / t_m)))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (-0.3333333333333333 * (((l / abs(k_m)) ^ 2.0) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(-0.3333333333333333 * N[(N[Power[N[(l / N[Abs[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. *-commutative 31.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-*l* 31.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-*l/ 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \color{blue}{\frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. +-commutative 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   6. unpow2 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   7. sqr-neg 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   8. distribute-frac-neg 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

   9. distribute-frac-neg 31.3%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

   10. unpow2 31.3%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

   11. associate--l+ 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

   12. metadata-eval 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

   14. unpow2 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   15. distribute-frac-neg 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

   16. distribute-frac-neg 41.4%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\left(-\frac{k}{t}\right) \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

3. Simplified 41.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 39.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

5. Taylor expanded in k around inf 33.3%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. *-commutative 33.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

   2. associate-/r* 33.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

7. Simplified 33.2%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. div-inv 33.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right)} \]

   2. pow-flip 33.4%

      \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \]

   3. metadata-eval 33.4%

      \[\leadsto -0.3333333333333333 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-2}}\right) \]

9. Applied egg-rr 33.4%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 31.8%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)\right)} \]

    2. expm1-udef 31.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-2}\right)} - 1\right)} \]

    3. *-commutative 31.4%

       \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-2} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

11. Applied egg-rr 31.4%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 31.8%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

    2. expm1-log1p 33.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

    3. associate-*r/ 33.3%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{k}^{-2} \cdot {\ell}^{2}}{t}} \]

    4. *-commutative 33.3%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-2}}}{t} \]

13. Simplified 34.7%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t}} \]

14. Final simplification 34.7%

    \[\leadsto -0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{\left|k\right|}\right)}^{2}}{t} \]

Reproduce

herbie shell --seed 2023332 
(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))))