Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 95.2%

Time: 23.5s

Alternatives: 11

Speedup: 2.0×

• 35.6% 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: 34.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.85 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k_m} \cdot \sqrt{\cos k_m}}{\sin k_m}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{{\sin k_m}^{2}}}{t_m}\\ \end{array} \end{array} \]

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 2.85e-18)
    (*
     2.0
     (pow (/ (/ (* (/ l k_m) (sqrt (cos k_m))) (sin k_m)) (sqrt t_m)) 2.0))
    (*
     2.0
     (/ (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (pow (sin 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 <= 2.85e-18) {
		tmp = 2.0 * pow(((((l / k_m) * sqrt(cos(k_m))) / sin(k_m)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / pow(sin(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 <= 2.85d-18) then
        tmp = 2.0d0 * (((((l / k_m) * sqrt(cos(k_m))) / sin(k_m)) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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 <= 2.85e-18) {
		tmp = 2.0 * Math.pow(((((l / k_m) * Math.sqrt(Math.cos(k_m))) / Math.sin(k_m)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / Math.pow(Math.sin(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 <= 2.85e-18:
		tmp = 2.0 * math.pow(((((l / k_m) * math.sqrt(math.cos(k_m))) / math.sin(k_m)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * ((l / k_m) * (l / k_m))) / math.pow(math.sin(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 <= 2.85e-18)
		tmp = Float64(2.0 * (Float64(Float64(Float64(Float64(l / k_m) * sqrt(cos(k_m))) / sin(k_m)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / (sin(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 <= 2.85e-18)
		tmp = 2.0 * (((((l / k_m) * sqrt(cos(k_m))) / sin(k_m)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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, 2.85e-18], N[(2.0 * N[Power[N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.84999999999999986e-18

   1. Initial program 36.9%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l/ 37.4%

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

      5. +-commutative 37.4%

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

      6. unpow2 37.4%

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

      7. sqr-neg 37.4%

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

      8. distribute-frac-neg 37.4%

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

      9. distribute-frac-neg 37.4%

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

      10. unpow2 37.4%

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

      11. associate--l+ 45.2%

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

      12. metadata-eval 45.2%

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

      13. +-rgt-identity 45.2%

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

      14. unpow2 45.2%

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

      15. distribute-frac-neg 45.2%

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

      16. distribute-frac-neg 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in k around inf 70.1%

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

      1. times-frac 70.6%

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

   6. Simplified 70.6%

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

      1. associate-*r/ 71.0%

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

      2. div-inv 71.0%

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

      3. pow-flip 71.0%

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

      4. metadata-eval 71.0%

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

   8. Applied egg-rr 71.0%

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

   9. Taylor expanded in l around 0 70.1%

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

       1. times-frac 70.6%

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

       2. associate-*r/ 71.0%

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

       3. unpow2 71.0%

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

       4. unpow2 71.0%

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

       5. times-frac 89.7%

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

       6. *-rgt-identity 89.7%

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

       7. associate-*r/ 89.7%

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

       8. *-rgt-identity 89.7%

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

       9. associate-*r/ 89.7%

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

       10. unpow-1 89.7%

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

       11. metadata-eval 89.7%

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

       12. unpow-1 89.7%

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

       13. metadata-eval 89.7%

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

       14. swap-sqr 71.0%

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

       15. unpow2 71.0%

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

       16. sqr-pow 71.0%

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

       17. *-commutative 71.0%

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

   11. Simplified 91.6%

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

       1. add-sqr-sqrt 47.7%

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

       2. pow2 47.7%

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

       3. sqrt-div 32.1%

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

       4. sqrt-div 33.0%

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

       5. *-commutative 33.0%

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

       6. sqrt-prod 28.9%

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

       7. unpow2 28.9%

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

       8. sqrt-prod 14.9%

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

       9. add-sqr-sqrt 30.4%

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

       10. unpow2 30.4%

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

       11. sqrt-prod 24.6%

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

       12. add-sqr-sqrt 31.3%

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

   13. Applied egg-rr 31.3%

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

   if 2.84999999999999986e-18 < k

   1. Initial program 22.9%

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

      1. associate-/r* 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*l* 22.9%

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

      4. associate-*l/ 22.9%

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

      5. +-commutative 22.9%

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

      6. unpow2 22.9%

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

      7. sqr-neg 22.9%

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

      8. distribute-frac-neg 22.9%

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

      9. distribute-frac-neg 22.9%

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

      10. unpow2 22.9%

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

      11. associate--l+ 34.2%

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

      12. metadata-eval 34.2%

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

      13. +-rgt-identity 34.2%

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

      14. unpow2 34.2%

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

      15. distribute-frac-neg 34.2%

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

      16. distribute-frac-neg 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in k around inf 67.5%

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

      1. times-frac 68.6%

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

   6. Simplified 68.6%

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

      1. associate-*r/ 68.6%

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

      2. div-inv 68.6%

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

      3. pow-flip 68.6%

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

      4. metadata-eval 68.6%

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

   8. Applied egg-rr 68.6%

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

   9. Taylor expanded in l around 0 67.5%

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

       1. times-frac 68.6%

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

       2. associate-*r/ 68.6%

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

       3. unpow2 68.6%

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

       4. unpow2 68.6%

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

       5. times-frac 90.7%

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

       6. *-rgt-identity 90.7%

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

       7. associate-*r/ 90.8%

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

       8. *-rgt-identity 90.8%

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

       9. associate-*r/ 90.8%

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

       10. unpow-1 90.8%

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

       11. metadata-eval 90.8%

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

       12. unpow-1 90.8%

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

       13. metadata-eval 90.8%

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

       14. swap-sqr 68.6%

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

       15. unpow2 68.6%

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

       16. sqr-pow 68.6%

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

       17. *-commutative 68.6%

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

   11. Simplified 90.8%

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

       1. unpow2 90.8%

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

   13. Applied egg-rr 90.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.85 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k} \cdot \sqrt{\cos k}}{\sin k}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 2: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot {\left(\frac{\sqrt{\cos k_m}}{\sqrt{t_m}} \cdot \frac{\frac{\ell}{k_m}}{\sin k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{{\sin k_m}^{2}}}{t_m}\\ \end{array} \end{array} \]

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 8.6e-18)
    (*
     2.0
     (pow (* (/ (sqrt (cos k_m)) (sqrt t_m)) (/ (/ l k_m) (sin k_m))) 2.0))
    (*
     2.0
     (/ (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (pow (sin 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 <= 8.6e-18) {
		tmp = 2.0 * pow(((sqrt(cos(k_m)) / sqrt(t_m)) * ((l / k_m) / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / pow(sin(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 <= 8.6d-18) then
        tmp = 2.0d0 * (((sqrt(cos(k_m)) / sqrt(t_m)) * ((l / k_m) / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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 <= 8.6e-18) {
		tmp = 2.0 * Math.pow(((Math.sqrt(Math.cos(k_m)) / Math.sqrt(t_m)) * ((l / k_m) / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / Math.pow(Math.sin(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 <= 8.6e-18:
		tmp = 2.0 * math.pow(((math.sqrt(math.cos(k_m)) / math.sqrt(t_m)) * ((l / k_m) / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * ((l / k_m) * (l / k_m))) / math.pow(math.sin(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 <= 8.6e-18)
		tmp = Float64(2.0 * (Float64(Float64(sqrt(cos(k_m)) / sqrt(t_m)) * Float64(Float64(l / k_m) / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / (sin(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 <= 8.6e-18)
		tmp = 2.0 * (((sqrt(cos(k_m)) / sqrt(t_m)) * ((l / k_m) / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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, 8.6e-18], N[(2.0 * N[Power[N[(N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.6000000000000005e-18

   1. Initial program 36.9%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l/ 37.4%

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

      5. +-commutative 37.4%

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

      6. unpow2 37.4%

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

      7. sqr-neg 37.4%

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

      8. distribute-frac-neg 37.4%

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

      9. distribute-frac-neg 37.4%

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

      10. unpow2 37.4%

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

      11. associate--l+ 45.2%

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

      12. metadata-eval 45.2%

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

      13. +-rgt-identity 45.2%

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

      14. unpow2 45.2%

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

      15. distribute-frac-neg 45.2%

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

      16. distribute-frac-neg 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in k around inf 70.1%

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

      1. times-frac 70.6%

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

   6. Simplified 70.6%

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

      1. associate-*r/ 71.0%

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

      2. div-inv 71.0%

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

      3. pow-flip 71.0%

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

      4. metadata-eval 71.0%

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

   8. Applied egg-rr 71.0%

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

   9. Taylor expanded in l around 0 70.1%

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

       1. times-frac 70.6%

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

       2. associate-*r/ 71.0%

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

       3. unpow2 71.0%

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

       4. unpow2 71.0%

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

       5. times-frac 89.7%

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

       6. *-rgt-identity 89.7%

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

       7. associate-*r/ 89.7%

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

       8. *-rgt-identity 89.7%

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

       9. associate-*r/ 89.7%

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

       10. unpow-1 89.7%

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

       11. metadata-eval 89.7%

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

       12. unpow-1 89.7%

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

       13. metadata-eval 89.7%

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

       14. swap-sqr 71.0%

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

       15. unpow2 71.0%

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

       16. sqr-pow 71.0%

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

       17. *-commutative 71.0%

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

   11. Simplified 91.6%

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

       1. add-sqr-sqrt 47.7%

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

       2. sqrt-div 32.1%

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

       3. sqrt-div 32.2%

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

       4. *-commutative 32.2%

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

       5. sqrt-prod 28.0%

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

       6. unpow2 28.0%

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

       7. sqrt-prod 13.9%

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

       8. add-sqr-sqrt 15.6%

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

       9. unpow2 15.6%

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

       10. sqrt-prod 10.9%

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

       11. add-sqr-sqrt 15.1%

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

       12. sqrt-div 15.1%

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

   13. Applied egg-rr 31.3%

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

       1. unpow2 31.3%

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

       2. associate-/l/ 31.3%

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

       3. *-commutative 31.3%

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

       4. times-frac 31.3%

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

   15. Simplified 31.3%

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

   if 8.6000000000000005e-18 < k

   1. Initial program 22.9%

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

      1. associate-/r* 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*l* 22.9%

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

      4. associate-*l/ 22.9%

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

      5. +-commutative 22.9%

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

      6. unpow2 22.9%

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

      7. sqr-neg 22.9%

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

      8. distribute-frac-neg 22.9%

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

      9. distribute-frac-neg 22.9%

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

      10. unpow2 22.9%

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

      11. associate--l+ 34.2%

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

      12. metadata-eval 34.2%

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

      13. +-rgt-identity 34.2%

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

      14. unpow2 34.2%

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

      15. distribute-frac-neg 34.2%

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

      16. distribute-frac-neg 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in k around inf 67.5%

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

      1. times-frac 68.6%

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

   6. Simplified 68.6%

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

      1. associate-*r/ 68.6%

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

      2. div-inv 68.6%

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

      3. pow-flip 68.6%

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

      4. metadata-eval 68.6%

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

   8. Applied egg-rr 68.6%

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

   9. Taylor expanded in l around 0 67.5%

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

       1. times-frac 68.6%

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

       2. associate-*r/ 68.6%

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

       3. unpow2 68.6%

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

       4. unpow2 68.6%

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

       5. times-frac 90.7%

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

       6. *-rgt-identity 90.7%

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

       7. associate-*r/ 90.8%

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

       8. *-rgt-identity 90.8%

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

       9. associate-*r/ 90.8%

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

       10. unpow-1 90.8%

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

       11. metadata-eval 90.8%

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

       12. unpow-1 90.8%

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

       13. metadata-eval 90.8%

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

       14. swap-sqr 68.6%

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

       15. unpow2 68.6%

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

       16. sqr-pow 68.6%

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

       17. *-commutative 68.6%

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

   11. Simplified 90.8%

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

       1. unpow2 90.8%

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

   13. Applied egg-rr 90.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot {\left(\frac{\sqrt{\cos k}}{\sqrt{t}} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 3: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.235:\\ \;\;\;\;\frac{2}{{\left(\frac{k_m}{\frac{\ell}{\sin k_m}} \cdot \sqrt{\frac{t_m}{\cos k_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{{\sin k_m}^{2}}}{t_m}\\ \end{array} \end{array} \]

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 0.235)
    (/ 2.0 (pow (* (/ k_m (/ l (sin k_m))) (sqrt (/ t_m (cos k_m)))) 2.0))
    (*
     2.0
     (/ (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (pow (sin 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 <= 0.235) {
		tmp = 2.0 / pow(((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / pow(sin(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 <= 0.235d0) then
        tmp = 2.0d0 / (((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))) ** 2.0d0)
    else
        tmp = 2.0d0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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 <= 0.235) {
		tmp = 2.0 / Math.pow(((k_m / (l / Math.sin(k_m))) * Math.sqrt((t_m / Math.cos(k_m)))), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / Math.pow(Math.sin(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 <= 0.235:
		tmp = 2.0 / math.pow(((k_m / (l / math.sin(k_m))) * math.sqrt((t_m / math.cos(k_m)))), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * ((l / k_m) * (l / k_m))) / math.pow(math.sin(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 <= 0.235)
		tmp = Float64(2.0 / (Float64(Float64(k_m / Float64(l / sin(k_m))) * sqrt(Float64(t_m / cos(k_m)))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / (sin(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 <= 0.235)
		tmp = 2.0 / (((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	else
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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, 0.235], N[(2.0 / N[Power[N[(N[(k$95$m / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.23499999999999999

   1. Initial program 36.4%

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

   3. Applied egg-rr 11.7%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2} + \left(\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot 0}} \]
   4. Step-by-step derivation

      1. mul0-rgt 24.8%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 24.8%

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

   5. Simplified 24.8%

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

   6. Taylor expanded in k around inf 40.5%

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

      1. associate-/l* 41.4%

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

   8. Simplified 41.4%

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

   if 0.23499999999999999 < k

   1. Initial program 24.1%

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

      1. associate-/r* 24.0%

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

      2. *-commutative 24.0%

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

      3. associate-*l* 24.0%

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

      4. associate-*l/ 24.0%

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

      5. +-commutative 24.0%

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

      6. unpow2 24.0%

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

      7. sqr-neg 24.0%

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

      8. distribute-frac-neg 24.0%

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

      9. distribute-frac-neg 24.0%

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

      10. unpow2 24.0%

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

      11. associate--l+ 36.0%

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

      12. metadata-eval 36.0%

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

      13. +-rgt-identity 36.0%

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

      14. unpow2 36.0%

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

      15. distribute-frac-neg 36.0%

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

      16. distribute-frac-neg 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in k around inf 65.9%

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

      1. times-frac 67.0%

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

   6. Simplified 67.0%

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

      1. associate-*r/ 67.0%

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

      2. div-inv 67.0%

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

      3. pow-flip 67.0%

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

      4. metadata-eval 67.0%

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

   8. Applied egg-rr 67.0%

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

   9. Taylor expanded in l around 0 65.9%

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

       1. times-frac 67.0%

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

       2. associate-*r/ 67.0%

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

       3. unpow2 67.0%

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

       4. unpow2 67.0%

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

       5. times-frac 90.3%

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

       6. *-rgt-identity 90.3%

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

       7. associate-*r/ 90.4%

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

       8. *-rgt-identity 90.4%

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

       9. associate-*r/ 90.4%

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

       10. unpow-1 90.4%

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

       11. metadata-eval 90.4%

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

       12. unpow-1 90.4%

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

       13. metadata-eval 90.4%

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

       14. swap-sqr 67.1%

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

       15. unpow2 67.1%

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

       16. sqr-pow 67.0%

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

       17. *-commutative 67.0%

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

   11. Simplified 90.3%

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

       1. unpow2 90.3%

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

   13. Applied egg-rr 90.3%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.235:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k_m}^{-2}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{{\sin k_m}^{2}}}{t_m}\\ \end{array} \end{array} \]

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 1.4e-14)
    (* 2.0 (pow (/ (* l (pow k_m -2.0)) (sqrt t_m)) 2.0))
    (*
     2.0
     (/ (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (pow (sin 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 <= 1.4e-14) {
		tmp = 2.0 * pow(((l * pow(k_m, -2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / pow(sin(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 <= 1.4d-14) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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 <= 1.4e-14) {
		tmp = 2.0 * Math.pow(((l * Math.pow(k_m, -2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / Math.pow(Math.sin(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 <= 1.4e-14:
		tmp = 2.0 * math.pow(((l * math.pow(k_m, -2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * ((l / k_m) * (l / k_m))) / math.pow(math.sin(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 <= 1.4e-14)
		tmp = Float64(2.0 * (Float64(Float64(l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / (sin(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 <= 1.4e-14)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (((cos(k_m) * ((l / k_m) * (l / k_m))) / (sin(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, 1.4e-14], N[(2.0 * N[Power[N[(N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-14

   1. Initial program 36.8%

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

      1. associate-/r* 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*l* 36.7%

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

      4. associate-*l/ 37.2%

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

      5. +-commutative 37.2%

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

      6. unpow2 37.2%

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

      7. sqr-neg 37.2%

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

      8. distribute-frac-neg 37.2%

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

      9. distribute-frac-neg 37.2%

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

      10. unpow2 37.2%

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

      11. associate--l+ 44.9%

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

      12. metadata-eval 44.9%

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

      13. +-rgt-identity 44.9%

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

      14. unpow2 44.9%

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

      15. distribute-frac-neg 44.9%

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

      16. distribute-frac-neg 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in k around 0 60.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 60.7%

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

      2. associate-/r* 57.4%

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

   6. Simplified 57.4%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. div-inv 57.4%

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

      2. pow-flip 57.4%

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

      3. metadata-eval 57.4%

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

   8. Applied egg-rr 57.4%

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

      1. add-sqr-sqrt 32.1%

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

      2. pow2 32.1%

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

      3. *-commutative 32.1%

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

      4. sqrt-prod 29.0%

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

      5. sqrt-pow1 31.5%

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

      6. metadata-eval 31.5%

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

      7. sqrt-div 24.0%

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

      8. unpow2 24.0%

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

      9. sqrt-prod 16.6%

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

      10. add-sqr-sqrt 31.1%

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

   10. Applied egg-rr 31.1%

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

       1. associate-*r/ 30.7%

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

   12. Simplified 30.7%

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

   if 1.4e-14 < k

   1. Initial program 23.3%

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

      1. associate-/r* 23.3%

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

      2. *-commutative 23.3%

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

      3. associate-*l* 23.2%

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

      4. associate-*l/ 23.2%

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

      5. +-commutative 23.2%

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

      6. unpow2 23.2%

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

      7. sqr-neg 23.2%

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

      8. distribute-frac-neg 23.2%

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

      9. distribute-frac-neg 23.2%

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

      10. unpow2 23.2%

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

      11. associate--l+ 34.8%

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

      12. metadata-eval 34.8%

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

      13. +-rgt-identity 34.8%

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

      14. unpow2 34.8%

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

      15. distribute-frac-neg 34.8%

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

      16. distribute-frac-neg 34.8%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in k around inf 67.0%

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

      1. times-frac 68.1%

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

   6. Simplified 68.1%

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

      1. associate-*r/ 68.1%

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

      2. div-inv 68.0%

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

      3. pow-flip 68.1%

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

      4. metadata-eval 68.1%

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

   8. Applied egg-rr 68.1%

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

   9. Taylor expanded in l around 0 67.0%

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

       1. times-frac 68.1%

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

       2. associate-*r/ 68.1%

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

       3. unpow2 68.1%

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

       4. unpow2 68.1%

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

       5. times-frac 90.6%

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

       6. *-rgt-identity 90.6%

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

       7. associate-*r/ 90.7%

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

       8. *-rgt-identity 90.7%

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

       9. associate-*r/ 90.7%

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

       10. unpow-1 90.7%

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

       11. metadata-eval 90.7%

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

       12. unpow-1 90.7%

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

       13. metadata-eval 90.7%

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

       14. swap-sqr 68.2%

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

       15. unpow2 68.2%

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

       16. sqr-pow 68.1%

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

       17. *-commutative 68.1%

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

   11. Simplified 90.6%

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

       1. unpow2 90.6%

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

   13. Applied egg-rr 90.6%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 5: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t_m \cdot \frac{2}{{\left(\frac{t_m}{\frac{\ell}{{k_m}^{1.5}}}\right)}^{2}}}{k_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left({k_m}^{-2} \cdot \frac{\ell}{\sqrt{t_m}}\right)}^{2}\\ \end{array} \end{array} \]

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 1.8e-144)
    (/ (* t_m (/ 2.0 (pow (/ t_m (/ l (pow k_m 1.5))) 2.0))) k_m)
    (* 2.0 (pow (* (pow k_m -2.0) (/ l (sqrt t_m))) 2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-144) {
		tmp = (t_m * (2.0 / pow((t_m / (l / pow(k_m, 1.5))), 2.0))) / k_m;
	} else {
		tmp = 2.0 * pow((pow(k_m, -2.0) * (l / sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-144) then
        tmp = (t_m * (2.0d0 / ((t_m / (l / (k_m ** 1.5d0))) ** 2.0d0))) / k_m
    else
        tmp = 2.0d0 * (((k_m ** (-2.0d0)) * (l / sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-144) {
		tmp = (t_m * (2.0 / Math.pow((t_m / (l / Math.pow(k_m, 1.5))), 2.0))) / k_m;
	} else {
		tmp = 2.0 * Math.pow((Math.pow(k_m, -2.0) * (l / Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.8e-144:
		tmp = (t_m * (2.0 / math.pow((t_m / (l / math.pow(k_m, 1.5))), 2.0))) / k_m
	else:
		tmp = 2.0 * math.pow((math.pow(k_m, -2.0) * (l / math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-144)
		tmp = Float64(Float64(t_m * Float64(2.0 / (Float64(t_m / Float64(l / (k_m ^ 1.5))) ^ 2.0))) / k_m);
	else
		tmp = Float64(2.0 * (Float64((k_m ^ -2.0) * Float64(l / sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-144)
		tmp = (t_m * (2.0 / ((t_m / (l / (k_m ^ 1.5))) ^ 2.0))) / k_m;
	else
		tmp = 2.0 * (((k_m ^ -2.0) * (l / sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.8e-144], N[(N[(t$95$m * N[(2.0 / N[Power[N[(t$95$m / N[(l / N[Power[k$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], N[(2.0 * N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-144

   1. Initial program 37.9%

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

      1. associate-/r* 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-*l* 37.9%

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

      4. associate-*l/ 37.9%

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

      5. +-commutative 37.9%

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

      6. unpow2 37.9%

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

      7. sqr-neg 37.9%

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

      8. distribute-frac-neg 37.9%

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

      9. distribute-frac-neg 37.9%

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

      10. unpow2 37.9%

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

      11. associate--l+ 46.6%

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

      12. metadata-eval 46.6%

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

      13. +-rgt-identity 46.6%

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

      14. unpow2 46.6%

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

      15. distribute-frac-neg 46.6%

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

      16. distribute-frac-neg 46.6%

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

   3. Simplified 46.6%

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

      1. *-un-lft-identity 46.6%

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

      2. unpow2 46.6%

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

      3. times-frac 49.8%

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

      4. clear-num 49.8%

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

   5. Applied egg-rr 49.9%

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

      1. associate-/l/ 49.9%

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

      2. associate-*r* 49.9%

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

      3. *-commutative 49.9%

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

   7. Simplified 49.9%

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

   8. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{k}^{3} \cdot {t}^{2}}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 49.6%

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

      2. associate-/l* 49.6%

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

   10. Simplified 49.6%

       \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{{\ell}^{2}}{{k}^{3}}}}} \]
   11. Step-by-step derivation

       1. associate-*l/ 49.6%

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

       2. add-sqr-sqrt 14.8%

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

       3. pow2 14.8%

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

       4. sqrt-div 35.4%

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

       5. unpow2 35.4%

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

       6. sqrt-prod 17.4%

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

       7. add-sqr-sqrt 43.6%

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

       8. sqrt-div 18.5%

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

       9. unpow2 18.5%

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

       10. sqrt-prod 14.9%

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

       11. add-sqr-sqrt 27.9%

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

       12. sqrt-pow1 29.7%

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

       13. metadata-eval 29.7%

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

   12. Applied egg-rr 29.7%

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

   if 1.8e-144 < k

   1. Initial program 26.0%

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

      1. associate-/r* 26.0%

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

      2. *-commutative 26.0%

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

      3. associate-*l* 25.9%

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

      4. associate-*l/ 27.0%

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

      5. +-commutative 27.0%

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

      6. unpow2 27.0%

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

      7. sqr-neg 27.0%

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

      8. distribute-frac-neg 27.0%

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

      9. distribute-frac-neg 27.0%

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

      10. unpow2 27.0%

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

      11. associate--l+ 35.5%

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

      12. metadata-eval 35.5%

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

      13. +-rgt-identity 35.5%

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

      14. unpow2 35.5%

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

      15. distribute-frac-neg 35.5%

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

      16. distribute-frac-neg 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in k around 0 58.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 58.5%

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

      2. associate-/r* 55.2%

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

   6. Simplified 55.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. div-inv 55.2%

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

      2. pow-flip 55.2%

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

      3. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

      3. *-commutative 43.2%

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

      4. sqrt-prod 34.6%

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

      5. sqrt-pow1 38.7%

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

      6. metadata-eval 38.7%

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

      7. sqrt-div 31.8%

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

      8. unpow2 31.8%

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

      9. sqrt-prod 18.3%

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

      10. add-sqr-sqrt 34.0%

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

   10. Applied egg-rr 34.0%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \frac{2}{{\left(\frac{t}{\frac{\ell}{{k}^{1.5}}}\right)}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left({k}^{-2} \cdot \frac{\ell}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 6: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-40}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k_m} \cdot \frac{t_m}{{\left({k_m}^{1.5} \cdot \frac{t_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= l 5e-40)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (* (/ 2.0 k_m) (/ t_m (pow (* (pow k_m 1.5) (/ t_m l)) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 5e-40) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = (2.0 / k_m) * (t_m / pow((pow(k_m, 1.5) * (t_m / l)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 5d-40) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = (2.0d0 / k_m) * (t_m / (((k_m ** 1.5d0) * (t_m / l)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 5e-40) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = (2.0 / k_m) * (t_m / Math.pow((Math.pow(k_m, 1.5) * (t_m / l)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 5e-40:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = (2.0 / k_m) * (t_m / math.pow((math.pow(k_m, 1.5) * (t_m / l)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 5e-40)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(Float64(2.0 / k_m) * Float64(t_m / (Float64((k_m ^ 1.5) * Float64(t_m / l)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 5e-40)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = (2.0 / k_m) * (t_m / (((k_m ^ 1.5) * (t_m / l)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 5e-40], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(t$95$m / N[Power[N[(N[Power[k$95$m, 1.5], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999965e-40

   1. Initial program 30.4%

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

      1. associate-/r* 30.3%

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

      2. *-commutative 30.3%

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

      3. associate-*l* 30.3%

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

      4. associate-*l/ 30.3%

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

      5. +-commutative 30.3%

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

      6. unpow2 30.3%

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

      7. sqr-neg 30.3%

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

      8. distribute-frac-neg 30.3%

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

      9. distribute-frac-neg 30.3%

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

      10. unpow2 30.3%

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

      11. associate--l+ 41.6%

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

      12. metadata-eval 41.6%

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

      13. +-rgt-identity 41.6%

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

      14. unpow2 41.6%

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

      15. distribute-frac-neg 41.6%

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

      16. distribute-frac-neg 41.6%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in k around 0 59.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 59.6%

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

      2. associate-/r* 56.2%

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

   6. Simplified 56.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. div-inv 56.2%

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

      2. pow-flip 56.2%

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

      3. metadata-eval 56.2%

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

   8. Applied egg-rr 56.2%

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

      1. associate-*l/ 59.6%

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

   10. Applied egg-rr 59.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 59.6%

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

       2. pow2 59.6%

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

       3. sqrt-prod 59.6%

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

       4. unpow2 59.6%

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

       5. sqrt-prod 26.2%

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

       6. add-sqr-sqrt 72.3%

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

       7. sqrt-pow1 76.8%

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

       8. metadata-eval 76.8%

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

   12. Applied egg-rr 76.8%

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

   if 4.99999999999999965e-40 < l

   1. Initial program 42.2%

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

      1. associate-/r* 42.2%

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

      2. *-commutative 42.2%

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

      3. associate-*l* 42.2%

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

      4. associate-*l/ 43.6%

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

      5. +-commutative 43.6%

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

      6. unpow2 43.6%

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

      7. sqr-neg 43.6%

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

      8. distribute-frac-neg 43.6%

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

      9. distribute-frac-neg 43.6%

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

      10. unpow2 43.6%

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

      11. associate--l+ 45.1%

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

      12. metadata-eval 45.1%

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

      13. +-rgt-identity 45.1%

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

      14. unpow2 45.1%

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

      15. distribute-frac-neg 45.1%

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

      16. distribute-frac-neg 45.1%

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

   3. Simplified 45.1%

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

      1. *-un-lft-identity 45.1%

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

      2. unpow2 45.1%

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

      3. times-frac 51.3%

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

      4. clear-num 51.3%

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

   5. Applied egg-rr 49.9%

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

      1. associate-/l/ 49.9%

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

      2. associate-*r* 49.9%

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

      3. *-commutative 49.9%

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

   7. Simplified 49.9%

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

   8. Taylor expanded in k around 0 51.2%

      \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{k}^{3} \cdot {t}^{2}}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 51.2%

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

      2. associate-/l* 51.2%

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

   10. Simplified 51.2%

       \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{{\ell}^{2}}{{k}^{3}}}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 23.6%

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

       2. expm1-udef 23.6%

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

   12. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{k} \cdot \frac{2}{{\left(\frac{t}{\frac{\ell}{{k}^{1.5}}}\right)}^{2}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 19.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{k} \cdot \frac{2}{{\left(\frac{t}{\frac{\ell}{{k}^{1.5}}}\right)}^{2}}\right)\right)} \]

       2. expm1-log1p 42.6%

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

       3. times-frac 44.0%

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

       4. *-commutative 44.0%

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

       5. times-frac 44.0%

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

       6. associate-/r/ 44.0%

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

   14. Simplified 44.0%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{t}{{\left(\frac{t}{\ell} \cdot {k}^{1.5}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-40}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{t}{{\left({k}^{1.5} \cdot \frac{t}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 7: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5000:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot t_m}{k_m \cdot {\left(\frac{t_m}{\frac{\ell}{{k_m}^{1.5}}}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= l 5000.0)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (/ (* 2.0 t_m) (* k_m (pow (/ t_m (/ l (pow k_m 1.5))) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 5000.0) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = (2.0 * t_m) / (k_m * pow((t_m / (l / pow(k_m, 1.5))), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 5000.0d0) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = (2.0d0 * t_m) / (k_m * ((t_m / (l / (k_m ** 1.5d0))) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 5000.0) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = (2.0 * t_m) / (k_m * Math.pow((t_m / (l / Math.pow(k_m, 1.5))), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 5000.0:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = (2.0 * t_m) / (k_m * math.pow((t_m / (l / math.pow(k_m, 1.5))), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 5000.0)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(Float64(2.0 * t_m) / Float64(k_m * (Float64(t_m / Float64(l / (k_m ^ 1.5))) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 5000.0)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = (2.0 * t_m) / (k_m * ((t_m / (l / (k_m ^ 1.5))) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 5000.0], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * t$95$m), $MachinePrecision] / N[(k$95$m * N[Power[N[(t$95$m / N[(l / N[Power[k$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5e3

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-*l* 31.0%

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

      4. associate-*l/ 31.5%

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

      5. +-commutative 31.5%

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

      6. unpow2 31.5%

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

      7. sqr-neg 31.5%

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

      8. distribute-frac-neg 31.5%

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

      9. distribute-frac-neg 31.5%

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

      10. unpow2 31.5%

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

      11. associate--l+ 42.1%

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

      12. metadata-eval 42.1%

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

      13. +-rgt-identity 42.1%

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

      14. unpow2 42.1%

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

      15. distribute-frac-neg 42.1%

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

      16. distribute-frac-neg 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in k around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 61.1%

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

      2. associate-/r* 57.9%

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

   6. Simplified 57.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. div-inv 57.9%

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

      2. pow-flip 57.9%

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

      3. metadata-eval 57.9%

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

   8. Applied egg-rr 57.9%

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

      1. associate-*l/ 61.1%

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

   10. Applied egg-rr 61.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 61.1%

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

       2. pow2 61.1%

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

       3. sqrt-prod 61.1%

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

       4. unpow2 61.1%

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

       5. sqrt-prod 29.7%

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

       6. add-sqr-sqrt 73.0%

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

       7. sqrt-pow1 77.2%

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

       8. metadata-eval 77.2%

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

   12. Applied egg-rr 77.2%

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

   if 5e3 < l

   1. Initial program 42.3%

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

      1. associate-/r* 42.3%

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

      2. *-commutative 42.3%

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

      3. associate-*l* 42.3%

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

      4. associate-*l/ 42.3%

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

      5. +-commutative 42.3%

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

      6. unpow2 42.3%

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

      7. sqr-neg 42.3%

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

      8. distribute-frac-neg 42.3%

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

      9. distribute-frac-neg 42.3%

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

      10. unpow2 42.3%

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

      11. associate--l+ 44.1%

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

      12. metadata-eval 44.1%

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

      13. +-rgt-identity 44.1%

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

      14. unpow2 44.1%

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

      15. distribute-frac-neg 44.1%

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

      16. distribute-frac-neg 44.1%

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

   3. Simplified 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. unpow2 44.1%

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

      3. times-frac 51.6%

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

      4. clear-num 51.6%

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

   5. Applied egg-rr 51.6%

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

      1. associate-/l/ 51.7%

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

      2. associate-*r* 51.7%

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

      3. *-commutative 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{k}^{3} \cdot {t}^{2}}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 49.6%

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

      2. associate-/l* 49.6%

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

   10. Simplified 49.6%

       \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{{\ell}^{2}}{{k}^{3}}}}} \]
   11. Step-by-step derivation

       1. frac-times 49.6%

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

       2. add-sqr-sqrt 36.9%

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

       3. pow2 36.9%

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

       4. sqrt-div 35.3%

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

       5. unpow2 35.3%

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

       6. sqrt-prod 14.1%

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

       7. add-sqr-sqrt 37.1%

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

       8. sqrt-div 37.0%

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

       9. unpow2 37.0%

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

       10. sqrt-prod 39.2%

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

       11. add-sqr-sqrt 39.2%

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

       12. sqrt-pow1 42.7%

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

       13. metadata-eval 42.7%

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

   12. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\frac{t \cdot 2}{k \cdot {\left(\frac{t}{\frac{\ell}{{k}^{1.5}}}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5000:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot t}{k \cdot {\left(\frac{t}{\frac{\ell}{{k}^{1.5}}}\right)}^{2}}\\ \end{array} \]

Alternative 8: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{t_m \cdot \frac{2}{{\left(\frac{t_m}{\frac{\ell}{{k_m}^{1.5}}}\right)}^{2}}}{k_m} \end{array} \]

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 (/ (* t_m (/ 2.0 (pow (/ t_m (/ l (pow k_m 1.5))) 2.0))) 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) {
	return t_s * ((t_m * (2.0 / pow((t_m / (l / pow(k_m, 1.5))), 2.0))) / k_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 * ((t_m * (2.0d0 / ((t_m / (l / (k_m ** 1.5d0))) ** 2.0d0))) / k_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 * ((t_m * (2.0 / Math.pow((t_m / (l / Math.pow(k_m, 1.5))), 2.0))) / k_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 * ((t_m * (2.0 / math.pow((t_m / (l / math.pow(k_m, 1.5))), 2.0))) / k_m)

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(t_m * Float64(2.0 / (Float64(t_m / Float64(l / (k_m ^ 1.5))) ^ 2.0))) / k_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 * ((t_m * (2.0 / ((t_m / (l / (k_m ^ 1.5))) ^ 2.0))) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(t$95$m * N[(2.0 / N[Power[N[(t$95$m / N[(l / N[Power[k$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

   1. associate-/r* 33.5%

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

   2. *-commutative 33.5%

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

   3. associate-*l* 33.5%

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

   4. associate-*l/ 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+ 42.5%

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

   12. metadata-eval 42.5%

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

   13. +-rgt-identity 42.5%

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

   14. unpow2 42.5%

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

   15. distribute-frac-neg 42.5%

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

   16. distribute-frac-neg 42.5%

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

3. Simplified 42.5%

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

   1. *-un-lft-identity 42.5%

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

   2. unpow2 42.5%

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

   3. times-frac 47.1%

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

   4. clear-num 47.1%

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

5. Applied egg-rr 47.4%

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

   1. associate-/l/ 47.4%

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

   2. associate-*r* 47.5%

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

   3. *-commutative 47.5%

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

7. Simplified 47.5%

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

8. Taylor expanded in k around 0 48.5%

   \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{k}^{3} \cdot {t}^{2}}{{\ell}^{2}}}} \]
9. Step-by-step derivation

   1. *-commutative 48.5%

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

   2. associate-/l* 48.9%

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

10. Simplified 48.9%

    \[\leadsto \frac{t}{k} \cdot \frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{{\ell}^{2}}{{k}^{3}}}}} \]
11. Step-by-step derivation

    1. associate-*l/ 48.9%

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

    2. add-sqr-sqrt 26.9%

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

    3. pow2 26.9%

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

    4. sqrt-div 39.9%

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

    5. unpow2 39.9%

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

    6. sqrt-prod 21.2%

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

    7. add-sqr-sqrt 47.9%

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

    8. sqrt-div 32.4%

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

    9. unpow2 32.4%

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

    10. sqrt-prod 20.4%

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

    11. add-sqr-sqrt 40.9%

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

    12. sqrt-pow1 43.1%

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

    13. metadata-eval 43.1%

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

12. Applied egg-rr 43.1%

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

13. Final simplification 43.1%

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

Alternative 9: 59.3% accurate, 2.0× speedup

Math

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

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 (* 2.0 (* (pow k_m -4.0) (/ (pow l 2.0) t_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. associate-/r* 33.5%

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

   2. *-commutative 33.5%

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

   3. associate-*l* 33.5%

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

   4. associate-*l/ 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+ 42.5%

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

   12. metadata-eval 42.5%

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

   13. +-rgt-identity 42.5%

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

   14. unpow2 42.5%

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

   15. distribute-frac-neg 42.5%

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

   16. distribute-frac-neg 42.5%

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

3. Simplified 42.5%

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

4. Taylor expanded in k around 0 58.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. *-commutative 58.5%

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

   2. associate-/r* 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. div-inv 56.1%

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

   2. pow-flip 56.1%

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

   3. metadata-eval 56.1%

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

8. Applied egg-rr 56.1%

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

9. Final simplification 56.1%

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

Alternative 10: 60.4% accurate, 2.0× speedup

Math

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

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 (* 2.0 (/ (* (pow l 2.0) (pow k_m -4.0)) t_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. associate-/r* 33.5%

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

   2. *-commutative 33.5%

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

   3. associate-*l* 33.5%

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

   4. associate-*l/ 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+ 42.5%

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

   12. metadata-eval 42.5%

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

   13. +-rgt-identity 42.5%

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

   14. unpow2 42.5%

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

   15. distribute-frac-neg 42.5%

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

   16. distribute-frac-neg 42.5%

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

3. Simplified 42.5%

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

4. Taylor expanded in k around 0 58.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. *-commutative 58.5%

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

   2. associate-/r* 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. div-inv 56.1%

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

   2. pow-flip 56.1%

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

   3. metadata-eval 56.1%

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

8. Applied egg-rr 56.1%

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

   1. associate-*l/ 58.6%

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

10. Applied egg-rr 58.6%

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

11. Final simplification 58.6%

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

Alternative 11: 69.9% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 33.5%

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

   1. associate-/r* 33.5%

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

   2. *-commutative 33.5%

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

   3. associate-*l* 33.5%

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

   4. associate-*l/ 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+ 42.5%

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

   12. metadata-eval 42.5%

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

   13. +-rgt-identity 42.5%

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

   14. unpow2 42.5%

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

   15. distribute-frac-neg 42.5%

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

   16. distribute-frac-neg 42.5%

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

3. Simplified 42.5%

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

4. Taylor expanded in k around 0 58.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. *-commutative 58.5%

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

   2. associate-/r* 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. div-inv 56.1%

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

   2. pow-flip 56.1%

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

   3. metadata-eval 56.1%

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

8. Applied egg-rr 56.1%

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

   1. associate-*l/ 58.6%

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

10. Applied egg-rr 58.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 58.6%

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

    2. pow2 58.6%

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

    3. sqrt-prod 58.6%

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

    4. unpow2 58.6%

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

    5. sqrt-prod 34.8%

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

    6. add-sqr-sqrt 68.4%

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

    7. sqrt-pow1 72.2%

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

    8. metadata-eval 72.2%

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

12. Applied egg-rr 72.2%

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

13. Final simplification 72.2%

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

Reproduce

herbie shell --seed 2023322 
(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))))