Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 95.0%

Time: 31.6s

Alternatives: 12

Speedup: 3.8×

• 35.1% 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: 34.9% 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: 95.0% accurate, 1.0× speedup

Math

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

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 3e-24)
    (* 2.0 (pow (* (/ k_m (/ l (sin k_m))) (sqrt (/ t_m (cos k_m)))) -2.0))
    (*
     2.0
     (/ (/ (* (cos k_m) (pow (/ l k_m) 2.0)) (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 <= 3e-24) {
		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) * pow((l / k_m), 2.0)) / 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 <= 3d-24) 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) ** 2.0d0)) / (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 <= 3e-24) {
		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) * Math.pow((l / k_m), 2.0)) / 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 <= 3e-24:
		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) * math.pow((l / k_m), 2.0)) / 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 <= 3e-24)
		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(l / k_m) ^ 2.0)) / (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 <= 3e-24)
		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) ^ 2.0)) / (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, 3e-24], 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[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $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 < 2.99999999999999995e-24

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate--l+ 34.1%

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

   3. Simplified 34.1%

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

      1. expm1-log1p-u 26.8%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 15.1%

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

   6. Applied egg-rr 20.2%

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

      1. expm1-def 22.2%

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

      2. expm1-log1p 22.3%

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

   8. Simplified 22.3%

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

   9. Taylor expanded in k around inf 40.6%

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

       1. associate-*l/ 40.6%

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

   11. Simplified 40.6%

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

       1. expm1-log1p-u 40.2%

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

       2. expm1-udef 35.1%

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

       3. div-inv 35.1%

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

       4. pow-flip 35.1%

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

       5. associate-/l* 35.1%

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

       6. metadata-eval 35.1%

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

   13. Applied egg-rr 35.1%

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

       1. expm1-def 40.6%

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

       2. expm1-log1p 40.9%

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

       3. associate-/r/ 40.9%

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

       4. associate-/l* 41.5%

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

   15. Simplified 41.5%

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

   if 2.99999999999999995e-24 < k

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-/r* 31.9%

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

      3. sub-neg 31.9%

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

      4. distribute-rgt-in 31.9%

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

      5. unpow2 31.9%

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

      6. times-frac 28.0%

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

      7. sqr-neg 28.0%

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

      8. times-frac 31.9%

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

      9. unpow2 31.9%

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

      10. distribute-rgt-in 31.9%

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

      11. +-commutative 31.9%

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

      12. associate-+l+ 37.8%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around 0 74.9%

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

      1. times-frac 76.1%

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

   7. Simplified 76.1%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-udef 62.5%

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

      3. div-inv 62.5%

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

      4. pow-flip 62.5%

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

      5. metadata-eval 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. expm1-def 70.5%

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

       2. expm1-log1p 76.1%

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

       3. associate-*r/ 76.2%

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

       4. *-commutative 76.2%

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

       5. associate-/r* 76.2%

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

       6. associate-*l* 76.2%

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

   11. Simplified 76.2%

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

       1. expm1-log1p-u 69.3%

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

       2. expm1-udef 58.8%

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

   13. Applied egg-rr 66.0%

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

       1. expm1-def 82.3%

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

       2. expm1-log1p 94.7%

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

       3. *-commutative 94.7%

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

   15. Simplified 94.7%

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

4. Final simplification 57.3%

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

Alternative 2: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k_m}^{2}}{\ell} \cdot \sqrt{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{\ell}{k_m} \cdot \sqrt{\cos k_m}}{\sin k_m}\right)}^{2}}{t_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e+175)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (* 2.0 (/ (pow (/ (* (/ l k_m) (sqrt (cos k_m))) (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 ((l * l) <= 2e+175) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (pow((((l / k_m) * sqrt(cos(k_m))) / 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 ((l * l) <= 2d+175) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((((l / k_m) * sqrt(cos(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 ((l * l) <= 2e+175) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((((l / k_m) * Math.sqrt(Math.cos(k_m))) / 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 (l * l) <= 2e+175:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * (math.pow((((l / k_m) * math.sqrt(math.cos(k_m))) / 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 (Float64(l * l) <= 2e+175)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(Float64(l / k_m) * sqrt(cos(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 ((l * l) <= 2e+175)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (((((l / k_m) * sqrt(cos(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[N[(l * l), $MachinePrecision], 2e+175], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[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], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.9999999999999999e175

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate--l+ 36.1%

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

   3. Simplified 36.1%

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

      1. expm1-log1p-u 23.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 20.0%

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

   6. Applied egg-rr 22.6%

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

      1. expm1-def 23.1%

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

      2. expm1-log1p 23.2%

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

   8. Simplified 23.2%

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

   9. Taylor expanded in k around 0 36.2%

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

   if 1.9999999999999999e175 < (*.f64 l l)

   1. Initial program 28.2%

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

      1. associate-*l* 28.2%

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

      2. associate-/r* 28.2%

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

      3. sub-neg 28.2%

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

      4. distribute-rgt-in 24.8%

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

      5. unpow2 24.8%

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

      6. times-frac 16.7%

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

      7. sqr-neg 16.7%

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

      8. times-frac 24.8%

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

      9. unpow2 24.8%

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

      10. distribute-rgt-in 28.2%

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

      11. +-commutative 28.2%

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

      12. associate-+l+ 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in t around 0 61.7%

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

      1. times-frac 65.1%

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

   7. Simplified 65.1%

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

      1. expm1-log1p-u 27.5%

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

      2. expm1-udef 24.3%

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

      3. div-inv 24.3%

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

      4. pow-flip 25.4%

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

      5. metadata-eval 25.4%

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

   9. Applied egg-rr 25.4%

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

       1. expm1-def 28.7%

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

       2. expm1-log1p 66.3%

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

       3. associate-*r/ 66.3%

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

       4. *-commutative 66.3%

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

       5. associate-/r* 65.3%

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

       6. associate-*l* 65.3%

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

   11. Simplified 65.3%

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

       1. add-sqr-sqrt 52.3%

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

       2. pow2 52.3%

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

   13. Applied egg-rr 67.8%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{\ell}{k} \cdot \sqrt{\cos k}}{\sin k}\right)}^{2}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k_m}^{2}}{\ell} \cdot \sqrt{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k_m}{\frac{\ell}{\sin k_m}} \cdot \sqrt{\frac{t_m}{\cos k_m}}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+147)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (* 2.0 (pow (* (/ k_m (/ l (sin k_m))) (sqrt (/ t_m (cos k_m)))) -2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+147) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * pow(((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+147) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+147) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * Math.pow(((k_m / (l / Math.sin(k_m))) * Math.sqrt((t_m / Math.cos(k_m)))), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+147:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * math.pow(((k_m / (l / math.sin(k_m))) * math.sqrt((t_m / math.cos(k_m)))), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+147)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * (Float64(Float64(k_m / Float64(l / sin(k_m))) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+147)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (((k_m / (l / sin(k_m))) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+147], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000002e147

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate--l+ 35.6%

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

   3. Simplified 35.6%

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

      1. expm1-log1p-u 23.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 19.8%

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

   6. Applied egg-rr 22.4%

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

      1. expm1-def 22.9%

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

      2. expm1-log1p 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in k around 0 36.2%

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

   if 5.0000000000000002e147 < (*.f64 l l)

   1. Initial program 29.5%

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

      1. associate-*l* 29.5%

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

      2. associate--l+ 29.5%

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

   3. Simplified 29.5%

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

      1. expm1-log1p-u 27.2%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 9.4%

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

   6. Applied egg-rr 16.8%

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

      1. expm1-def 19.9%

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

      2. expm1-log1p 20.0%

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

   8. Simplified 20.0%

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

   9. Taylor expanded in k around inf 46.3%

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

       1. associate-*l/ 46.3%

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

   11. Simplified 46.3%

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

       1. expm1-log1p-u 45.1%

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

       2. expm1-udef 37.8%

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

       3. div-inv 37.8%

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

       4. pow-flip 37.8%

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

       5. associate-/l* 36.8%

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

       6. metadata-eval 36.8%

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

   13. Applied egg-rr 36.8%

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

       1. expm1-def 44.2%

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

       2. expm1-log1p 45.3%

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

       3. associate-/r/ 46.3%

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

       4. associate-/l* 46.3%

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

   15. Simplified 46.3%

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

4. Final simplification 39.7%

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

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{k_m}{\frac{\ell}{\sin k_m}} \cdot \sqrt{\frac{t_m}{\cos k_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\cos k_m \cdot {\left(\frac{\ell}{k_m}\right)}^{2}\right) \cdot {\sin k_m}^{-2}}{t_m}\\ \end{array} \end{array} \]

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-24)
    (* 2.0 (pow (* (/ k_m (/ l (sin k_m))) (sqrt (/ t_m (cos k_m)))) -2.0))
    (*
     2.0
     (/ (* (* (cos k_m) (pow (/ l k_m) 2.0)) (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-24) {
		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) * pow((l / k_m), 2.0)) * 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-24) 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) ** 2.0d0)) * (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-24) {
		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) * Math.pow((l / k_m), 2.0)) * 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-24:
		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) * math.pow((l / k_m), 2.0)) * 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-24)
		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(l / k_m) ^ 2.0)) * (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-24)
		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) ^ 2.0)) * (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-24], 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[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $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 < 1.4000000000000001e-24

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate--l+ 34.1%

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

   3. Simplified 34.1%

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

      1. expm1-log1p-u 26.8%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 15.1%

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

   6. Applied egg-rr 20.2%

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

      1. expm1-def 22.2%

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

      2. expm1-log1p 22.3%

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

   8. Simplified 22.3%

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

   9. Taylor expanded in k around inf 40.6%

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

       1. associate-*l/ 40.6%

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

   11. Simplified 40.6%

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

       1. expm1-log1p-u 40.2%

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

       2. expm1-udef 35.1%

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

       3. div-inv 35.1%

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

       4. pow-flip 35.1%

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

       5. associate-/l* 35.1%

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

       6. metadata-eval 35.1%

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

   13. Applied egg-rr 35.1%

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

       1. expm1-def 40.6%

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

       2. expm1-log1p 40.9%

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

       3. associate-/r/ 40.9%

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

       4. associate-/l* 41.5%

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

   15. Simplified 41.5%

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

   if 1.4000000000000001e-24 < k

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-/r* 31.9%

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

      3. sub-neg 31.9%

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

      4. distribute-rgt-in 31.9%

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

      5. unpow2 31.9%

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

      6. times-frac 28.0%

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

      7. sqr-neg 28.0%

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

      8. times-frac 31.9%

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

      9. unpow2 31.9%

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

      10. distribute-rgt-in 31.9%

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

      11. +-commutative 31.9%

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

      12. associate-+l+ 37.8%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around 0 74.9%

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

      1. times-frac 76.1%

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

   7. Simplified 76.1%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-udef 62.5%

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

      3. div-inv 62.5%

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

      4. pow-flip 62.5%

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

      5. metadata-eval 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. expm1-def 70.5%

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

       2. expm1-log1p 76.1%

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

       3. associate-*r/ 76.2%

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

       4. *-commutative 76.2%

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

       5. associate-/r* 76.2%

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

       6. associate-*l* 76.2%

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

   11. Simplified 76.2%

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

       1. div-inv 76.1%

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

       2. metadata-eval 76.1%

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

       3. pow-flip 76.2%

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

       4. associate-*r* 76.2%

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

       5. div-inv 76.1%

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

       6. add-sqr-sqrt 76.1%

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

       7. pow2 76.1%

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

       8. sqrt-div 76.1%

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

       9. unpow2 76.1%

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

       10. sqrt-prod 50.1%

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

       11. add-sqr-sqrt 84.7%

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

       12. unpow2 84.7%

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

       13. sqrt-prod 94.5%

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

       14. add-sqr-sqrt 94.6%

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

       15. pow-flip 94.6%

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

       16. metadata-eval 94.6%

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

   13. Applied egg-rr 94.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}\right) \cdot {\sin k}^{-2}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{{\sin k_m}^{2}}}{t_m}\\ \mathbf{elif}\;t_m \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{2}{\frac{\frac{k_m}{t_m}}{\frac{t_m}{k_m}} \cdot \left(\left(\sin k_m \cdot \tan k_m\right) \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k_m}^{2} \cdot \sqrt{t_m}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= t_m 5.5e-93)
    (* 2.0 (/ (/ (pow (/ l k_m) 2.0) (pow (sin k_m) 2.0)) t_m))
    (if (<= t_m 7.2e+74)
      (/
       2.0
       (*
        (/ (/ k_m t_m) (/ t_m k_m))
        (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/ 2.0 (pow (/ (* (pow k_m 2.0) (sqrt 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 (t_m <= 5.5e-93) {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / pow(sin(k_m), 2.0)) / t_m);
	} else if (t_m <= 7.2e+74) {
		tmp = 2.0 / (((k_m / t_m) / (t_m / k_m)) * ((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / pow(((pow(k_m, 2.0) * sqrt(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 (t_m <= 5.5d-93) then
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / (sin(k_m) ** 2.0d0)) / t_m)
    else if (t_m <= 7.2d+74) then
        tmp = 2.0d0 / (((k_m / t_m) / (t_m / k_m)) * ((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    else
        tmp = 2.0d0 / ((((k_m ** 2.0d0) * sqrt(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 (t_m <= 5.5e-93) {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / Math.pow(Math.sin(k_m), 2.0)) / t_m);
	} else if (t_m <= 7.2e+74) {
		tmp = 2.0 / (((k_m / t_m) / (t_m / k_m)) * ((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) * Math.sqrt(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 t_m <= 5.5e-93:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / math.pow(math.sin(k_m), 2.0)) / t_m)
	elif t_m <= 7.2e+74:
		tmp = 2.0 / (((k_m / t_m) / (t_m / k_m)) * ((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	else:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) * math.sqrt(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 (t_m <= 5.5e-93)
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / (sin(k_m) ^ 2.0)) / t_m));
	elseif (t_m <= 7.2e+74)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t_m) / Float64(t_m / k_m)) * Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) * sqrt(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 (t_m <= 5.5e-93)
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / (sin(k_m) ^ 2.0)) / t_m);
	elseif (t_m <= 7.2e+74)
		tmp = 2.0 / (((k_m / t_m) / (t_m / k_m)) * ((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l) * (t_m / l))));
	else
		tmp = 2.0 / ((((k_m ^ 2.0) * sqrt(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[t$95$m, 5.5e-93], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+74], N[(2.0 / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.49999999999999968e-93

   1. Initial program 28.4%

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

      1. associate-*l* 28.4%

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

      2. associate-/r* 28.4%

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

      3. sub-neg 28.4%

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

      4. distribute-rgt-in 25.1%

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

      5. unpow2 25.1%

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

      6. times-frac 14.0%

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

      7. sqr-neg 14.0%

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

      8. times-frac 25.1%

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

      9. unpow2 25.1%

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

      10. distribute-rgt-in 28.4%

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

      11. +-commutative 28.4%

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

      12. associate-+l+ 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in t around 0 75.9%

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

      1. times-frac 76.5%

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

   7. Simplified 76.5%

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

      1. expm1-log1p-u 48.5%

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

      2. expm1-udef 43.8%

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

      3. div-inv 43.8%

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

      4. pow-flip 44.4%

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

      5. metadata-eval 44.4%

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

   9. Applied egg-rr 44.4%

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

       1. expm1-def 49.1%

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

       2. expm1-log1p 76.6%

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

       3. associate-*r/ 76.6%

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

       4. *-commutative 76.6%

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

       5. associate-/r* 77.2%

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

       6. associate-*l* 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in k around 0 64.3%

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

       1. unpow2 64.3%

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

       2. unpow2 64.3%

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

       3. times-frac 70.0%

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

       4. unpow2 70.0%

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

   14. Simplified 70.0%

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

   if 5.49999999999999968e-93 < t < 7.19999999999999975e74

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate--l+ 77.5%

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

   3. Simplified 77.5%

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

      1. associate-+r- 77.5%

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

      2. add-exp-log 76.0%

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

      3. log1p-udef 76.0%

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

      4. expm1-udef 77.8%

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

      5. expm1-log1p-u 79.3%

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

      6. unpow2 79.3%

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

      7. clear-num 79.3%

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

      8. un-div-inv 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. unpow3 79.2%

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

      2. times-frac 85.4%

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

      3. pow2 85.4%

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

   8. Applied egg-rr 85.4%

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

   if 7.19999999999999975e74 < t

   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-*l* 22.9%

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

      2. associate--l+ 22.9%

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

   3. Simplified 22.9%

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

      1. expm1-log1p-u 13.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 13.9%

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

   6. Applied egg-rr 47.5%

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

      1. expm1-def 51.5%

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

      2. expm1-log1p 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in k around inf 67.2%

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

       1. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

   12. Taylor expanded in k around 0 83.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}}{t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{{\sin k_m}^{2}}}{t_m}\\ \mathbf{elif}\;t_m \leq 7 \cdot 10^{+74}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t_m}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k_m \cdot \tan k_m\right)\right) \cdot \frac{\frac{k_m}{t_m}}{\frac{t_m}{k_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k_m}^{2} \cdot \sqrt{t_m}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= t_m 7.5e-93)
    (* 2.0 (/ (/ (pow (/ l k_m) 2.0) (pow (sin k_m) 2.0)) t_m))
    (if (<= t_m 7e+74)
      (/
       2.0
       (*
        (* (* (/ (pow t_m 3.0) l) (/ 1.0 l)) (* (sin k_m) (tan k_m)))
        (/ (/ k_m t_m) (/ t_m k_m))))
      (/ 2.0 (pow (/ (* (pow k_m 2.0) (sqrt 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 (t_m <= 7.5e-93) {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / pow(sin(k_m), 2.0)) / t_m);
	} else if (t_m <= 7e+74) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = 2.0 / pow(((pow(k_m, 2.0) * sqrt(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 (t_m <= 7.5d-93) then
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / (sin(k_m) ** 2.0d0)) / t_m)
    else if (t_m <= 7d+74) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l) * (1.0d0 / l)) * (sin(k_m) * tan(k_m))) * ((k_m / t_m) / (t_m / k_m)))
    else
        tmp = 2.0d0 / ((((k_m ** 2.0d0) * sqrt(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 (t_m <= 7.5e-93) {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / Math.pow(Math.sin(k_m), 2.0)) / t_m);
	} else if (t_m <= 7e+74) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) * (1.0 / l)) * (Math.sin(k_m) * Math.tan(k_m))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) * Math.sqrt(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 t_m <= 7.5e-93:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / math.pow(math.sin(k_m), 2.0)) / t_m)
	elif t_m <= 7e+74:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l) * (1.0 / l)) * (math.sin(k_m) * math.tan(k_m))) * ((k_m / t_m) / (t_m / k_m)))
	else:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) * math.sqrt(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 (t_m <= 7.5e-93)
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / (sin(k_m) ^ 2.0)) / t_m));
	elseif (t_m <= 7e+74)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(1.0 / l)) * Float64(sin(k_m) * tan(k_m))) * Float64(Float64(k_m / t_m) / Float64(t_m / k_m))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) * sqrt(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 (t_m <= 7.5e-93)
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / (sin(k_m) ^ 2.0)) / t_m);
	elseif (t_m <= 7e+74)
		tmp = 2.0 / (((((t_m ^ 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m))) * ((k_m / t_m) / (t_m / k_m)));
	else
		tmp = 2.0 / ((((k_m ^ 2.0) * sqrt(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[t$95$m, 7.5e-93], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+74], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.50000000000000034e-93

   1. Initial program 28.4%

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

      1. associate-*l* 28.4%

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

      2. associate-/r* 28.4%

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

      3. sub-neg 28.4%

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

      4. distribute-rgt-in 25.1%

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

      5. unpow2 25.1%

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

      6. times-frac 14.0%

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

      7. sqr-neg 14.0%

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

      8. times-frac 25.1%

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

      9. unpow2 25.1%

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

      10. distribute-rgt-in 28.4%

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

      11. +-commutative 28.4%

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

      12. associate-+l+ 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in t around 0 75.9%

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

      1. times-frac 76.5%

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

   7. Simplified 76.5%

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

      1. expm1-log1p-u 48.5%

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

      2. expm1-udef 43.8%

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

      3. div-inv 43.8%

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

      4. pow-flip 44.4%

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

      5. metadata-eval 44.4%

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

   9. Applied egg-rr 44.4%

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

       1. expm1-def 49.1%

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

       2. expm1-log1p 76.6%

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

       3. associate-*r/ 76.6%

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

       4. *-commutative 76.6%

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

       5. associate-/r* 77.2%

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

       6. associate-*l* 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in k around 0 64.3%

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

       1. unpow2 64.3%

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

       2. unpow2 64.3%

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

       3. times-frac 70.0%

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

       4. unpow2 70.0%

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

   14. Simplified 70.0%

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

   if 7.50000000000000034e-93 < t < 7.00000000000000029e74

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate--l+ 77.5%

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

   3. Simplified 77.5%

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

      1. associate-+r- 77.5%

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

      2. add-exp-log 76.0%

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

      3. log1p-udef 76.0%

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

      4. expm1-udef 77.8%

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

      5. expm1-log1p-u 79.3%

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

      6. unpow2 79.3%

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

      7. clear-num 79.3%

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

      8. un-div-inv 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-/r* 85.5%

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

      2. div-inv 85.6%

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

   8. Applied egg-rr 85.6%

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

   if 7.00000000000000029e74 < t

   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-*l* 22.9%

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

      2. associate--l+ 22.9%

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

   3. Simplified 22.9%

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

      1. expm1-log1p-u 13.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 13.9%

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

   6. Applied egg-rr 47.5%

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

      1. expm1-def 51.5%

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

      2. expm1-log1p 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in k around inf 67.2%

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

       1. associate-*l/ 67.2%

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

   11. Simplified 67.2%

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

   12. Taylor expanded in k around 0 83.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}}{t}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+74}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 1.3× speedup

Math

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

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 (<= (* l l) 1e+204)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (* 2.0 (/ (/ (pow (/ l k_m) 2.0) (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 ((l * l) <= 1e+204) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / 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 ((l * l) <= 1d+204) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / (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 ((l * l) <= 1e+204) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / 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 (l * l) <= 1e+204:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / 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 (Float64(l * l) <= 1e+204)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / (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 ((l * l) <= 1e+204)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / (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[N[(l * l), $MachinePrecision], 1e+204], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $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 (*.f64 l l) < 9.99999999999999989e203

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate--l+ 36.1%

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

   3. Simplified 36.1%

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

      1. expm1-log1p-u 23.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 20.3%

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

   6. Applied egg-rr 22.8%

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

      1. expm1-def 23.9%

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

      2. expm1-log1p 24.0%

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

   8. Simplified 24.0%

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

   9. Taylor expanded in k around 0 36.7%

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

   if 9.99999999999999989e203 < (*.f64 l l)

   1. Initial program 27.9%

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

      1. associate-*l* 27.9%

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

      2. associate-/r* 27.9%

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

      3. sub-neg 27.9%

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

      4. distribute-rgt-in 24.3%

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

      5. unpow2 24.3%

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

      6. times-frac 16.0%

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

      7. sqr-neg 16.0%

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

      8. times-frac 24.3%

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

      9. unpow2 24.3%

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

      10. distribute-rgt-in 27.9%

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

      11. +-commutative 27.9%

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

      12. associate-+l+ 28.0%

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

   3. Simplified 28.0%

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

   5. Taylor expanded in t around 0 61.5%

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

      1. times-frac 63.9%

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

   7. Simplified 63.9%

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

      1. expm1-log1p-u 25.1%

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

      2. expm1-udef 22.8%

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

      3. div-inv 22.8%

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

      4. pow-flip 24.0%

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

      5. metadata-eval 24.0%

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

   9. Applied egg-rr 24.0%

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

       1. expm1-def 26.3%

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

       2. expm1-log1p 65.1%

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

       3. associate-*r/ 65.1%

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

       4. *-commutative 65.1%

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

       5. associate-/r* 65.1%

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

       6. associate-*l* 65.1%

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

   11. Simplified 65.1%

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

   12. Taylor expanded in k around 0 49.0%

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

       1. unpow2 49.0%

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

       2. unpow2 49.0%

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

       3. times-frac 55.7%

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

       4. unpow2 55.7%

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

   14. Simplified 55.7%

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

4. Final simplification 42.9%

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

Alternative 8: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k_m}^{2}}{\ell} \cdot \sqrt{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k_m}\right)}^{2} \cdot \frac{{k_m}^{-2} + -0.16666666666666666}{t_m}\right)\\ \end{array} \end{array} \]

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 5.4e-101)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (*
     2.0
     (*
      (pow (/ l k_m) 2.0)
      (/ (+ (pow k_m -2.0) -0.16666666666666666) 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 (l <= 5.4e-101) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((pow(k_m, -2.0) + -0.16666666666666666) / 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 (l <= 5.4d-101) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * (((k_m ** (-2.0d0)) + (-0.16666666666666666d0)) / 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 (l <= 5.4e-101) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.pow(k_m, -2.0) + -0.16666666666666666) / 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 l <= 5.4e-101:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.pow(k_m, -2.0) + -0.16666666666666666) / 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 (l <= 5.4e-101)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64((k_m ^ -2.0) + -0.16666666666666666) / 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 (l <= 5.4e-101)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (((k_m ^ -2.0) + -0.16666666666666666) / 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[l, 5.4e-101], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Power[k$95$m, -2.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.4000000000000003e-101

   1. Initial program 31.8%

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

      1. associate-*l* 31.8%

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

      2. associate--l+ 31.8%

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

   3. Simplified 31.8%

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

      1. expm1-log1p-u 22.1%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)\right)}} \]

      2. expm1-udef 16.4%

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

   6. Applied egg-rr 20.1%

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

      1. expm1-def 21.6%

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

      2. expm1-log1p 21.8%

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

   8. Simplified 21.8%

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

   9. Taylor expanded in k around 0 32.8%

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

   if 5.4000000000000003e-101 < l

   1. Initial program 37.1%

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

      1. associate-*l* 37.1%

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

      2. associate-/r* 37.1%

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

      3. sub-neg 37.1%

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

      4. distribute-rgt-in 34.5%

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

      5. unpow2 34.5%

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

      6. times-frac 23.1%

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

      7. sqr-neg 23.1%

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

      8. times-frac 34.5%

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

      9. unpow2 34.5%

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

      10. distribute-rgt-in 37.1%

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

      11. +-commutative 37.1%

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

      12. associate-+l+ 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around 0 77.1%

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

      1. times-frac 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in k around 0 58.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* 58.5%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. associate-*r/ 58.5%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      3. metadata-eval 58.5%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   10. Simplified 58.5%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

   11. Taylor expanded in t around 0 58.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. times-frac 58.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. unpow2 58.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       3. unpow2 58.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       4. times-frac 60.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       5. unpow2 60.7%

          \[\leadsto 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       6. sub-neg 60.7%

          \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       7. unpow-1 60.7%

          \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{{\left({k}^{2}\right)}^{-1}} + \left(-0.16666666666666666\right)}{t}\right) \]

       8. exp-to-pow 43.2%

          \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{\color{blue}{\left(e^{\log k \cdot 2}\right)}}^{-1} + \left(-0.16666666666666666\right)}{t}\right) \]

       9. *-commutative 43.2%

          \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{\left(e^{\color{blue}{2 \cdot \log k}}\right)}^{-1} + \left(-0.16666666666666666\right)}{t}\right) \]

       10. exp-prod 43.2%

           \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{e^{\left(2 \cdot \log k\right) \cdot -1}} + \left(-0.16666666666666666\right)}{t}\right) \]

       11. *-commutative 43.2%

           \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{e^{\color{blue}{-1 \cdot \left(2 \cdot \log k\right)}} + \left(-0.16666666666666666\right)}{t}\right) \]

       12. associate-*r* 43.2%

           \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{e^{\color{blue}{\left(-1 \cdot 2\right) \cdot \log k}} + \left(-0.16666666666666666\right)}{t}\right) \]

       13. metadata-eval 43.2%

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

       14. *-commutative 43.2%

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

       15. exp-to-pow 60.7%

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

       16. metadata-eval 60.7%

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

   13. Simplified 60.7%

       \[\leadsto 2 \cdot \color{blue}{\left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{k}^{-2} + -0.16666666666666666}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{k}^{-2} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{t_m \cdot {k_m}^{4}} \cdot {\ell}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k_m}\right)}^{2} \cdot \frac{-0.16666666666666666}{t_m}\right)\\ \end{array} \end{array} \]

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 7.2e+49)
    (* (/ 2.0 (* t_m (pow k_m 4.0))) (pow l 2.0))
    (* 2.0 (* (pow (/ l k_m) 2.0) (/ -0.16666666666666666 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 <= 7.2e+49) {
		tmp = (2.0 / (t_m * pow(k_m, 4.0))) * pow(l, 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2d+49) then
        tmp = (2.0d0 / (t_m * (k_m ** 4.0d0))) * (l ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((-0.16666666666666666d0) / 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 <= 7.2e+49) {
		tmp = (2.0 / (t_m * Math.pow(k_m, 4.0))) * Math.pow(l, 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2e+49:
		tmp = (2.0 / (t_m * math.pow(k_m, 4.0))) * math.pow(l, 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2e+49)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k_m ^ 4.0))) * (l ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(-0.16666666666666666 / 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 <= 7.2e+49)
		tmp = (2.0 / (t_m * (k_m ^ 4.0))) * (l ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (-0.16666666666666666 / 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, 7.2e+49], N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999993e49

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate--l+ 32.4%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in k around 0 64.3%

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

      1. associate-/r/ 64.3%

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

      2. *-commutative 64.3%

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

   7. Applied egg-rr 64.3%

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

   if 7.19999999999999993e49 < k

   1. Initial program 37.2%

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

      1. associate-*l* 37.2%

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

      2. associate-/r* 37.2%

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

      3. sub-neg 37.2%

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

      4. distribute-rgt-in 37.2%

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

      5. unpow2 37.2%

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

      6. times-frac 31.9%

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

      7. sqr-neg 31.9%

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

      8. times-frac 37.2%

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

      9. unpow2 37.2%

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

      10. distribute-rgt-in 37.2%

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

      11. +-commutative 37.2%

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

      12. associate-+l+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 74.2%

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

      1. times-frac 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in k around 0 63.3%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. associate-*r/ 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      3. metadata-eval 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   10. Simplified 63.3%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

   11. Taylor expanded in k around inf 63.2%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 63.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. *-commutative 63.2%

          \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

       3. times-frac 63.3%

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

       4. unpow2 63.3%

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

       5. unpow2 63.3%

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

       6. times-frac 66.6%

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

       7. unpow2 66.6%

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

   13. Simplified 66.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{\left(t_m \cdot {k_m}^{4}\right) \cdot {\ell}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k_m}\right)}^{2} \cdot \frac{-0.16666666666666666}{t_m}\right)\\ \end{array} \end{array} \]

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 7.2e+49)
    (/ 2.0 (* (* t_m (pow k_m 4.0)) (pow l -2.0)))
    (* 2.0 (* (pow (/ l k_m) 2.0) (/ -0.16666666666666666 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 <= 7.2e+49) {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) * pow(l, -2.0));
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2d+49) then
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) * (l ** (-2.0d0)))
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((-0.16666666666666666d0) / 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 <= 7.2e+49) {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) * Math.pow(l, -2.0));
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2e+49:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) * math.pow(l, -2.0))
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (-0.16666666666666666 / 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 <= 7.2e+49)
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) * (l ^ -2.0)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(-0.16666666666666666 / 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 <= 7.2e+49)
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) * (l ^ -2.0));
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (-0.16666666666666666 / 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, 7.2e+49], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999993e49

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate--l+ 32.4%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in k around 0 64.3%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)\right)}} \]

      2. expm1-udef 25.8%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)} - 1}} \]

      3. div-inv 25.8%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{4} \cdot t\right) \cdot \frac{1}{{\ell}^{2}}}\right)} - 1} \]

      4. *-commutative 25.8%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left(t \cdot {k}^{4}\right)} \cdot \frac{1}{{\ell}^{2}}\right)} - 1} \]

      5. pow-flip 25.8%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)} - 1} \]

      6. metadata-eval 25.8%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{\color{blue}{-2}}\right)} - 1} \]

   7. Applied egg-rr 25.8%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 46.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}\right)\right)}} \]

      2. expm1-log1p 64.4%

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

   9. Simplified 64.4%

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

   if 7.19999999999999993e49 < k

   1. Initial program 37.2%

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

      1. associate-*l* 37.2%

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

      2. associate-/r* 37.2%

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

      3. sub-neg 37.2%

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

      4. distribute-rgt-in 37.2%

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

      5. unpow2 37.2%

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

      6. times-frac 31.9%

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

      7. sqr-neg 31.9%

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

      8. times-frac 37.2%

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

      9. unpow2 37.2%

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

      10. distribute-rgt-in 37.2%

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

      11. +-commutative 37.2%

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

      12. associate-+l+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 74.2%

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

      1. times-frac 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in k around 0 63.3%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. associate-*r/ 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      3. metadata-eval 63.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   10. Simplified 63.3%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

   11. Taylor expanded in k around inf 63.2%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 63.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. *-commutative 63.2%

          \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

       3. times-frac 63.3%

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

       4. unpow2 63.3%

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

       5. unpow2 63.3%

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

       6. times-frac 66.6%

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

       7. unpow2 66.6%

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

   13. Simplified 66.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 2.0× speedup

Math

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

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 (/ l k_m) 2.0) (/ (+ (pow k_m -2.0) -0.16666666666666666) 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 / k_m), 2.0) * ((pow(k_m, -2.0) + -0.16666666666666666) / 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) * (((k_m ** (-2.0d0)) + (-0.16666666666666666d0)) / 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 / k_m), 2.0) * ((Math.pow(k_m, -2.0) + -0.16666666666666666) / 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 / k_m), 2.0) * ((math.pow(k_m, -2.0) + -0.16666666666666666) / 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) * Float64(Float64((k_m ^ -2.0) + -0.16666666666666666) / 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) * (((k_m ^ -2.0) + -0.16666666666666666) / 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 / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Power[k$95$m, -2.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. associate-*l* 33.4%

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

   2. associate-/r* 33.4%

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

   3. sub-neg 33.4%

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

   4. distribute-rgt-in 31.1%

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

   5. unpow2 31.1%

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

   6. times-frac 22.1%

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

   7. sqr-neg 22.1%

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

   8. times-frac 31.1%

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

   9. unpow2 31.1%

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

   10. distribute-rgt-in 33.4%

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

   11. +-commutative 33.4%

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

   12. associate-+l+ 41.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in t around 0 76.2%

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

   1. times-frac 77.4%

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

7. Simplified 77.4%

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

8. Taylor expanded in k around 0 64.8%

   \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
9. Step-by-step derivation

   1. associate-/r* 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   2. associate-*r/ 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   3. metadata-eval 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

10. Simplified 64.5%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

11. Taylor expanded in t around 0 64.3%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. times-frac 64.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

    2. unpow2 64.5%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    3. unpow2 64.5%

       \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    4. times-frac 71.4%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    5. unpow2 71.4%

       \[\leadsto 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    6. sub-neg 71.4%

       \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

    7. unpow-1 71.4%

       \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{{\left({k}^{2}\right)}^{-1}} + \left(-0.16666666666666666\right)}{t}\right) \]

    8. exp-to-pow 42.4%

       \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{\color{blue}{\left(e^{\log k \cdot 2}\right)}}^{-1} + \left(-0.16666666666666666\right)}{t}\right) \]

    9. *-commutative 42.4%

       \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{{\left(e^{\color{blue}{2 \cdot \log k}}\right)}^{-1} + \left(-0.16666666666666666\right)}{t}\right) \]

    10. exp-prod 42.4%

        \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{e^{\left(2 \cdot \log k\right) \cdot -1}} + \left(-0.16666666666666666\right)}{t}\right) \]

    11. *-commutative 42.4%

        \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{e^{\color{blue}{-1 \cdot \left(2 \cdot \log k\right)}} + \left(-0.16666666666666666\right)}{t}\right) \]

    12. associate-*r* 42.4%

        \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{e^{\color{blue}{\left(-1 \cdot 2\right) \cdot \log k}} + \left(-0.16666666666666666\right)}{t}\right) \]

    13. metadata-eval 42.4%

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

    14. *-commutative 42.4%

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

    15. exp-to-pow 71.4%

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

    16. metadata-eval 71.4%

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

13. Simplified 71.4%

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

14. Final simplification 71.4%

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

Alternative 12: 34.9% accurate, 3.8× speedup

Math

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

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 (/ l k_m) 2.0) (/ -0.16666666666666666 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 / k_m), 2.0) * (-0.16666666666666666 / 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) * ((-0.16666666666666666d0) / 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 / k_m), 2.0) * (-0.16666666666666666 / 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 / k_m), 2.0) * (-0.16666666666666666 / 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) * Float64(-0.16666666666666666 / 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) * (-0.16666666666666666 / 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 / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. associate-*l* 33.4%

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

   2. associate-/r* 33.4%

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

   3. sub-neg 33.4%

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

   4. distribute-rgt-in 31.1%

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

   5. unpow2 31.1%

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

   6. times-frac 22.1%

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

   7. sqr-neg 22.1%

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

   8. times-frac 31.1%

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

   9. unpow2 31.1%

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

   10. distribute-rgt-in 33.4%

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

   11. +-commutative 33.4%

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

   12. associate-+l+ 41.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in t around 0 76.2%

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

   1. times-frac 77.4%

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

7. Simplified 77.4%

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

8. Taylor expanded in k around 0 64.8%

   \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
9. Step-by-step derivation

   1. associate-/r* 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   2. associate-*r/ 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   3. metadata-eval 64.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

10. Simplified 64.5%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

11. Taylor expanded in k around inf 37.1%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 37.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 37.1%

       \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    3. times-frac 37.2%

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

    4. unpow2 37.2%

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

    5. unpow2 37.2%

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

    6. times-frac 39.4%

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

    7. unpow2 39.4%

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

13. Simplified 39.4%

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

14. Final simplification 39.4%

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

Reproduce

herbie shell --seed 2024024 
(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))))