Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 80.7%

Time: 17.7s

Alternatives: 19

Speedup: 24.8×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2} \cdot \tan k\_m\right) \cdot \left(1 + \left(1 + \frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \mathsf{fma}\left(2, \frac{{\left(\frac{t\_m}{\ell}\right)}^{2} \cdot t\_2}{\cos k\_m}, \frac{t\_2}{{\ell}^{2}} \cdot \frac{{k\_m}^{2}}{\cos k\_m}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= k_m 6.2e-133)
      (/
       2.0
       (*
        (* (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k_m))) 2.0) (tan k_m))
        (+ 1.0 (+ 1.0 (* (/ k_m t_m) (/ k_m t_m))))))
      (/
       2.0
       (*
        t_m
        (fma
         2.0
         (/ (* (pow (/ t_m l) 2.0) t_2) (cos k_m))
         (* (/ t_2 (pow l 2.0)) (/ (pow k_m 2.0) (cos k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 6.2e-133) {
		tmp = 2.0 / ((pow(((pow(t_m, 1.5) / l) * sqrt(sin(k_m))), 2.0) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	} else {
		tmp = 2.0 / (t_m * fma(2.0, ((pow((t_m / l), 2.0) * t_2) / cos(k_m)), ((t_2 / pow(l, 2.0)) * (pow(k_m, 2.0) / cos(k_m)))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 6.2e-133)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0) * tan(k_m)) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) * Float64(k_m / t_m))))));
	else
		tmp = Float64(2.0 / Float64(t_m * fma(2.0, Float64(Float64((Float64(t_m / l) ^ 2.0) * t_2) / cos(k_m)), Float64(Float64(t_2 / (l ^ 2.0)) * Float64((k_m ^ 2.0) / cos(k_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.2e-133], N[(2.0 / N[(N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(2.0 * N[(N[(N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000032e-133

   1. Initial program 61.2%

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

      1. add-sqr-sqrt 32.4%

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

      2. pow2 32.4%

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

      3. associate-/r* 37.1%

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

      4. *-commutative 37.1%

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

      5. sqrt-prod 17.4%

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

      6. associate-/r* 15.0%

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

      7. sqrt-div 15.0%

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

      8. sqrt-pow1 16.2%

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

      9. metadata-eval 16.2%

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

      10. sqrt-prod 10.7%

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

      11. add-sqr-sqrt 19.2%

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

   4. Applied egg-rr 19.2%

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

      1. *-commutative 19.2%

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

   6. Simplified 19.2%

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

      1. unpow2 19.2%

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

   8. Applied egg-rr 19.2%

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

   if 6.20000000000000032e-133 < k

   1. Initial program 57.4%

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

      1. add-sqr-sqrt 31.1%

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

      2. pow2 31.1%

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

      3. associate-/r* 37.1%

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

      4. *-commutative 37.1%

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

      5. sqrt-prod 23.4%

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

      6. associate-/r* 19.6%

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

      7. sqrt-div 19.6%

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

      8. sqrt-pow1 21.8%

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

      9. metadata-eval 21.8%

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

      10. sqrt-prod 17.0%

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

      11. add-sqr-sqrt 26.2%

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

   4. Applied egg-rr 26.2%

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

      1. *-commutative 26.2%

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

   6. Simplified 26.2%

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

      1. div-inv 26.2%

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

      2. associate-*l* 26.2%

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

      3. *-commutative 26.2%

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

      4. unpow-prod-down 26.1%

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

      5. pow2 26.1%

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

      6. add-sqr-sqrt 33.1%

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

      7. add-sqr-sqrt 33.1%

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

      8. pow2 33.1%

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

   8. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 33.1%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 32.0%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 32.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 32.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 32.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 32.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 32.0%

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

   11. Taylor expanded in t around 0 73.1%

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

       1. fma-define 73.1%

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

       2. times-frac 73.1%

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

       3. associate-*r/ 73.1%

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

       4. unpow2 73.1%

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

       5. unpow2 73.1%

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

       6. times-frac 80.9%

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

       7. unpow1 80.9%

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

       8. pow-plus 80.9%

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

       9. metadata-eval 80.9%

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

       10. *-commutative 80.9%

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

       11. times-frac 82.1%

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

   13. Simplified 82.1%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(2, \frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot {\sin k}^{2}}{\cos k}, \frac{{\sin k}^{2}}{{\ell}^{2}} \cdot \frac{{k}^{2}}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{2}{\left({\left(t\_2 \cdot \sqrt{\sin k\_m}\right)}^{2} \cdot \tan k\_m\right) \cdot \left(1 + \left(1 + \frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right) \cdot \left(t\_2 \cdot \sqrt{\sin k\_m \cdot \tan k\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= k_m 2.4e-111)
      (/
       2.0
       (*
        (* (pow (* t_2 (sqrt (sin k_m))) 2.0) (tan k_m))
        (+ 1.0 (+ 1.0 (* (/ k_m t_m) (/ k_m t_m))))))
      (if (<= k_m 1.95e-5)
        (pow
         (/
          (sqrt 2.0)
          (*
           (hypot 1.0 (hypot 1.0 (/ k_m t_m)))
           (* t_2 (sqrt (* (sin k_m) (tan k_m))))))
         2.0)
        (/
         2.0
         (*
          (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
          (/ (pow (sin k_m) 2.0) (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 2.4e-111) {
		tmp = 2.0 / ((pow((t_2 * sqrt(sin(k_m))), 2.0) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	} else if (k_m <= 1.95e-5) {
		tmp = pow((sqrt(2.0) / (hypot(1.0, hypot(1.0, (k_m / t_m))) * (t_2 * sqrt((sin(k_m) * tan(k_m)))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 2.4e-111) {
		tmp = 2.0 / ((Math.pow((t_2 * Math.sqrt(Math.sin(k_m))), 2.0) * Math.tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	} else if (k_m <= 1.95e-5) {
		tmp = Math.pow((Math.sqrt(2.0) / (Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))) * (t_2 * Math.sqrt((Math.sin(k_m) * Math.tan(k_m)))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if k_m <= 2.4e-111:
		tmp = 2.0 / ((math.pow((t_2 * math.sqrt(math.sin(k_m))), 2.0) * math.tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))))
	elif k_m <= 1.95e-5:
		tmp = math.pow((math.sqrt(2.0) / (math.hypot(1.0, math.hypot(1.0, (k_m / t_m))) * (t_2 * math.sqrt((math.sin(k_m) * math.tan(k_m)))))), 2.0)
	else:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k_m <= 2.4e-111)
		tmp = Float64(2.0 / Float64(Float64((Float64(t_2 * sqrt(sin(k_m))) ^ 2.0) * tan(k_m)) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) * Float64(k_m / t_m))))));
	elseif (k_m <= 1.95e-5)
		tmp = Float64(sqrt(2.0) / Float64(hypot(1.0, hypot(1.0, Float64(k_m / t_m))) * Float64(t_2 * sqrt(Float64(sin(k_m) * tan(k_m)))))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (k_m <= 2.4e-111)
		tmp = 2.0 / ((((t_2 * sqrt(sin(k_m))) ^ 2.0) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	elseif (k_m <= 1.95e-5)
		tmp = (sqrt(2.0) / (hypot(1.0, hypot(1.0, (k_m / t_m))) * (t_2 * sqrt((sin(k_m) * tan(k_m)))))) ^ 2.0;
	else
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.4e-111], N[(2.0 / N[(N[(N[Power[N[(t$95$2 * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.95e-5], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.4000000000000001e-111

   1. Initial program 61.5%

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

      1. add-sqr-sqrt 32.8%

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

      2. pow2 32.8%

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

      3. associate-/r* 37.4%

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

      4. *-commutative 37.4%

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

      5. sqrt-prod 18.2%

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

      6. associate-/r* 15.9%

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

      7. sqrt-div 15.8%

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

      8. sqrt-pow1 17.0%

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

      9. metadata-eval 17.0%

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

      10. sqrt-prod 11.1%

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

      11. add-sqr-sqrt 19.9%

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

   4. Applied egg-rr 19.9%

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

      1. *-commutative 19.9%

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

   6. Simplified 19.9%

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

      1. unpow2 19.9%

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

   8. Applied egg-rr 19.9%

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

   if 2.4000000000000001e-111 < k < 1.95e-5

   1. Initial program 63.6%

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

      1. add-sqr-sqrt 42.9%

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

   4. Applied egg-rr 37.3%

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

      1. unpow2 37.3%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}} \]

   6. Simplified 37.3%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}} \]

   if 1.95e-5 < k

   1. Initial program 53.8%

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

   3. Simplified 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. *-un-lft-identity 58.7%

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

      3. times-frac 58.7%

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

      4. pow2 58.7%

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

      5. cbrt-div 58.7%

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

      6. rem-cbrt-cube 58.7%

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

      7. cbrt-div 58.7%

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

      8. rem-cbrt-cube 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

   8. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. cube-prod 66.8%

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

      3. rem-cube-cbrt 66.8%

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

      4. associate-*l* 66.8%

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

   10. Simplified 66.8%

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

   11. Taylor expanded in k around inf 76.8%

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

       1. associate-*r* 76.8%

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

       2. times-frac 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\left({\left(t\_2 \cdot \sqrt{\sin k\_m}\right)}^{2} \cdot \tan k\_m\right) \cdot \left(1 + \left(1 + \frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right) \cdot \left(t\_2 \cdot \sqrt{\sin k\_m \cdot \tan k\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= k_m 1.3e-101)
      (/
       2.0
       (*
        (* (pow (* t_2 (sqrt (sin k_m))) 2.0) (tan k_m))
        (+ 1.0 (+ 1.0 (* (/ k_m t_m) (/ k_m t_m))))))
      (if (<= k_m 1.65e-5)
        (/
         2.0
         (pow
          (*
           (hypot 1.0 (hypot 1.0 (/ k_m t_m)))
           (* t_2 (sqrt (* (sin k_m) (tan k_m)))))
          2.0))
        (/
         2.0
         (*
          (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
          (/ (pow (sin k_m) 2.0) (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 1.3e-101) {
		tmp = 2.0 / ((pow((t_2 * sqrt(sin(k_m))), 2.0) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	} else if (k_m <= 1.65e-5) {
		tmp = 2.0 / pow((hypot(1.0, hypot(1.0, (k_m / t_m))) * (t_2 * sqrt((sin(k_m) * tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 1.3e-101) {
		tmp = 2.0 / ((Math.pow((t_2 * Math.sqrt(Math.sin(k_m))), 2.0) * Math.tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	} else if (k_m <= 1.65e-5) {
		tmp = 2.0 / Math.pow((Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))) * (t_2 * Math.sqrt((Math.sin(k_m) * Math.tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if k_m <= 1.3e-101:
		tmp = 2.0 / ((math.pow((t_2 * math.sqrt(math.sin(k_m))), 2.0) * math.tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))))
	elif k_m <= 1.65e-5:
		tmp = 2.0 / math.pow((math.hypot(1.0, math.hypot(1.0, (k_m / t_m))) * (t_2 * math.sqrt((math.sin(k_m) * math.tan(k_m))))), 2.0)
	else:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k_m <= 1.3e-101)
		tmp = Float64(2.0 / Float64(Float64((Float64(t_2 * sqrt(sin(k_m))) ^ 2.0) * tan(k_m)) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) * Float64(k_m / t_m))))));
	elseif (k_m <= 1.65e-5)
		tmp = Float64(2.0 / (Float64(hypot(1.0, hypot(1.0, Float64(k_m / t_m))) * Float64(t_2 * sqrt(Float64(sin(k_m) * tan(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (k_m <= 1.3e-101)
		tmp = 2.0 / ((((t_2 * sqrt(sin(k_m))) ^ 2.0) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))));
	elseif (k_m <= 1.65e-5)
		tmp = 2.0 / ((hypot(1.0, hypot(1.0, (k_m / t_m))) * (t_2 * sqrt((sin(k_m) * tan(k_m))))) ^ 2.0);
	else
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.3e-101], N[(2.0 / N[(N[(N[Power[N[(t$95$2 * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.65e-5], N[(2.0 / N[Power[N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3000000000000001e-101

   1. Initial program 61.5%

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

      1. add-sqr-sqrt 32.8%

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

      2. pow2 32.8%

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

      3. associate-/r* 37.4%

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

      4. *-commutative 37.4%

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

      5. sqrt-prod 18.2%

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

      6. associate-/r* 15.9%

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

      7. sqrt-div 15.8%

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

      8. sqrt-pow1 17.0%

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

      9. metadata-eval 17.0%

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

      10. sqrt-prod 11.1%

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

      11. add-sqr-sqrt 19.9%

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

   4. Applied egg-rr 19.9%

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

      1. *-commutative 19.9%

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

   6. Simplified 19.9%

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

      1. unpow2 19.9%

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

   8. Applied egg-rr 19.9%

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

   if 1.3000000000000001e-101 < k < 1.6500000000000001e-5

   1. Initial program 63.6%

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

      1. div-inv 63.6%

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

      2. add-sqr-sqrt 21.8%

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

      3. pow2 21.8%

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

   4. Applied egg-rr 37.3%

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

      1. associate-*r/ 37.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 37.3%

         \[\leadsto \frac{\color{blue}{2}}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}} \]

   6. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{2}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   if 1.6500000000000001e-5 < k

   1. Initial program 53.8%

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

   3. Simplified 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. *-un-lft-identity 58.7%

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

      3. times-frac 58.7%

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

      4. pow2 58.7%

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

      5. cbrt-div 58.7%

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

      6. rem-cbrt-cube 58.7%

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

      7. cbrt-div 58.7%

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

      8. rem-cbrt-cube 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

   8. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. cube-prod 66.8%

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

      3. rem-cube-cbrt 66.8%

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

      4. associate-*l* 66.8%

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

   10. Simplified 66.8%

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

   11. Taylor expanded in k around inf 76.8%

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

       1. associate-*r* 76.8%

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

       2. times-frac 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k\_m}}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}}{\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.7e-98)
    (/
     2.0
     (*
      (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
      (/ (pow (sin k_m) 2.0) (cos k_m))))
    (/
     (/ (/ 2.0 (sin k_m)) (pow (/ (pow t_m 1.5) l) 2.0))
     (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.7e-98) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	} else {
		tmp = ((2.0 / sin(k_m)) / pow((pow(t_m, 1.5) / l), 2.0)) / (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.7d-98) then
        tmp = 2.0d0 / (((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)) * ((sin(k_m) ** 2.0d0) / cos(k_m)))
    else
        tmp = ((2.0d0 / sin(k_m)) / (((t_m ** 1.5d0) / l) ** 2.0d0)) / (tan(k_m) * (2.0d0 + ((k_m / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.7e-98) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = ((2.0 / Math.sin(k_m)) / Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) / (Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4.7e-98:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	else:
		tmp = ((2.0 / math.sin(k_m)) / math.pow((math.pow(t_m, 1.5) / l), 2.0)) / (math.tan(k_m) * (2.0 + math.pow((k_m / t_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4.7e-98)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(Float64(Float64(2.0 / sin(k_m)) / (Float64((t_m ^ 1.5) / l) ^ 2.0)) / Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4.7e-98)
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	else
		tmp = ((2.0 / sin(k_m)) / (((t_m ^ 1.5) / l) ^ 2.0)) / (tan(k_m) * (2.0 + ((k_m / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.7e-98], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.70000000000000005e-98

   1. Initial program 54.1%

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

   3. Simplified 56.2%

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

      1. add-cube-cbrt 56.1%

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

      2. *-un-lft-identity 56.1%

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

      3. times-frac 56.1%

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

      4. pow2 56.1%

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

      5. cbrt-div 56.2%

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

      6. rem-cbrt-cube 56.2%

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

      7. cbrt-div 56.1%

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

      8. rem-cbrt-cube 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. add-cube-cbrt 65.2%

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

      2. pow3 65.2%

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

   8. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. cube-prod 66.6%

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

      3. rem-cube-cbrt 66.6%

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

      4. associate-*l* 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in k around inf 65.9%

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

       1. associate-*r* 65.9%

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

       2. times-frac 67.8%

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

   13. Simplified 67.8%

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

   if 4.70000000000000005e-98 < t

   1. Initial program 70.2%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. associate-/r* 44.9%

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

      4. *-commutative 44.9%

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

      5. sqrt-prod 45.0%

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

      6. associate-/r* 39.5%

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

      7. sqrt-div 39.5%

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

      8. sqrt-pow1 39.5%

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

      9. metadata-eval 39.5%

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

      10. sqrt-prod 26.1%

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

      11. add-sqr-sqrt 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

   6. Simplified 46.0%

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

      1. div-inv 46.0%

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

      2. associate-*l* 46.0%

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

      3. *-commutative 46.0%

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

      4. unpow-prod-down 44.9%

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

      5. pow2 44.9%

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

      6. add-sqr-sqrt 80.0%

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

      7. add-sqr-sqrt 80.0%

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

      8. pow2 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. associate-*r/ 80.0%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 80.0%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 79.9%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 80.0%

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

       1. div-inv 80.0%

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

       2. associate-*r* 80.0%

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

       3. pow2 80.0%

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

       4. associate-+r+ 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow2 80.0%

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

   12. Applied egg-rr 80.0%

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

       1. associate-*r/ 80.0%

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

       2. metadata-eval 80.0%

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

       3. associate-/r* 80.0%

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

       4. associate-/r* 80.0%

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

       5. *-commutative 80.0%

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

   14. Simplified 80.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left({\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-100)
    (/
     2.0
     (*
      (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
      (/ (pow (sin k_m) 2.0) (cos k_m))))
    (/
     2.0
     (*
      (sin k_m)
      (*
       (pow (/ (pow t_m 1.5) l) 2.0)
       (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 6.8e-100) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	} else {
		tmp = 2.0 / (sin(k_m) * (pow((pow(t_m, 1.5) / l), 2.0) * (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 6.8d-100) then
        tmp = 2.0d0 / (((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)) * ((sin(k_m) ** 2.0d0) / cos(k_m)))
    else
        tmp = 2.0d0 / (sin(k_m) * ((((t_m ** 1.5d0) / l) ** 2.0d0) * (tan(k_m) * (2.0d0 + ((k_m / t_m) ** 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 6.8e-100) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 6.8e-100:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	else:
		tmp = 2.0 / (math.sin(k_m) * (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (math.tan(k_m) * (2.0 + math.pow((k_m / t_m), 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 6.8e-100)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 6.8e-100)
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	else
		tmp = 2.0 / (sin(k_m) * ((((t_m ^ 1.5) / l) ^ 2.0) * (tan(k_m) * (2.0 + ((k_m / t_m) ^ 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-100], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.79999999999999953e-100

   1. Initial program 54.1%

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

   3. Simplified 56.2%

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

      1. add-cube-cbrt 56.1%

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

      2. *-un-lft-identity 56.1%

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

      3. times-frac 56.1%

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

      4. pow2 56.1%

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

      5. cbrt-div 56.2%

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

      6. rem-cbrt-cube 56.2%

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

      7. cbrt-div 56.1%

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

      8. rem-cbrt-cube 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. add-cube-cbrt 65.2%

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

      2. pow3 65.2%

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

   8. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. cube-prod 66.6%

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

      3. rem-cube-cbrt 66.6%

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

      4. associate-*l* 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in k around inf 65.9%

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

       1. associate-*r* 65.9%

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

       2. times-frac 67.8%

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

   13. Simplified 67.8%

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

   if 6.79999999999999953e-100 < t

   1. Initial program 70.2%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. associate-/r* 44.9%

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

      4. *-commutative 44.9%

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

      5. sqrt-prod 45.0%

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

      6. associate-/r* 39.5%

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

      7. sqrt-div 39.5%

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

      8. sqrt-pow1 39.5%

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

      9. metadata-eval 39.5%

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

      10. sqrt-prod 26.1%

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

      11. add-sqr-sqrt 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

   6. Simplified 46.0%

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

      1. div-inv 46.0%

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

      2. associate-*l* 46.0%

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

      3. *-commutative 46.0%

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

      4. unpow-prod-down 44.9%

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

      5. pow2 44.9%

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

      6. add-sqr-sqrt 80.0%

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

      7. add-sqr-sqrt 80.0%

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

      8. pow2 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. associate-*r/ 80.0%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 80.0%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 79.9%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 79.9%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 80.0%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 80.0%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 80.0%

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

       1. associate-*r* 80.0%

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

       2. add-sqr-sqrt 44.9%

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

       3. pow2 44.9%

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

       4. unpow-prod-down 46.0%

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

       5. *-commutative 46.0%

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

       6. +-commutative 46.0%

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

       7. pow2 46.0%

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

       8. associate-*l* 46.0%

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

       9. distribute-lft-in 46.0%

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

   12. Applied egg-rr 80.0%

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

       1. distribute-lft-out 80.0%

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

       2. *-commutative 80.0%

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

       3. +-commutative 80.0%

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

       4. unpow2 80.0%

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

       5. times-frac 69.0%

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

       6. unpow2 69.0%

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

       7. unpow2 69.0%

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

       8. associate-+l+ 69.0%

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

       9. metadata-eval 69.0%

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

       10. +-commutative 69.0%

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

       11. unpow2 69.0%

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

       12. unpow2 69.0%

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

       13. times-frac 80.0%

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

       14. unpow2 80.0%

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

       15. associate-*r* 80.0%

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

       16. associate-*l* 80.0%

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

   14. Simplified 80.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + \frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-99)
    (/
     2.0
     (*
      (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
      (/ (pow (sin k_m) 2.0) (cos k_m))))
    (/
     2.0
     (*
      (+ 1.0 (+ 1.0 (* (/ k_m t_m) (/ k_m t_m))))
      (* (tan k_m) (* (sin k_m) (pow (/ (pow t_m 1.5) 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 <= 3.2e-99) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * pow((pow(t_m, 1.5) / 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 <= 3.2d-99) then
        tmp = 2.0d0 / (((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)) * ((sin(k_m) ** 2.0d0) / cos(k_m)))
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * (((t_m ** 1.5d0) / 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 <= 3.2e-99) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (Math.tan(k_m) * (Math.sin(k_m) * Math.pow((Math.pow(t_m, 1.5) / 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 <= 3.2e-99:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	else:
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (math.tan(k_m) * (math.sin(k_m) * math.pow((math.pow(t_m, 1.5) / 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 <= 3.2e-99)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) * Float64(k_m / t_m)))) * Float64(tan(k_m) * Float64(sin(k_m) * (Float64((t_m ^ 1.5) / 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 <= 3.2e-99)
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * (((t_m ^ 1.5) / 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, 3.2e-99], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.2000000000000001e-99

   1. Initial program 54.1%

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

   3. Simplified 56.2%

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

      1. add-cube-cbrt 56.1%

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

      2. *-un-lft-identity 56.1%

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

      3. times-frac 56.1%

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

      4. pow2 56.1%

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

      5. cbrt-div 56.2%

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

      6. rem-cbrt-cube 56.2%

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

      7. cbrt-div 56.1%

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

      8. rem-cbrt-cube 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. add-cube-cbrt 65.2%

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

      2. pow3 65.2%

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

   8. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. cube-prod 66.6%

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

      3. rem-cube-cbrt 66.6%

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

      4. associate-*l* 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in k around inf 65.9%

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

       1. associate-*r* 65.9%

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

       2. times-frac 67.8%

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

   13. Simplified 67.8%

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

   if 3.2000000000000001e-99 < t

   1. Initial program 70.2%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. associate-/r* 44.9%

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

      4. *-commutative 44.9%

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

      5. sqrt-prod 45.0%

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

      6. associate-/r* 39.5%

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

      7. sqrt-div 39.5%

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

      8. sqrt-pow1 39.5%

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

      9. metadata-eval 39.5%

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

      10. sqrt-prod 26.1%

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

      11. add-sqr-sqrt 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

   6. Simplified 46.0%

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

      1. unpow2 46.0%

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

   8. Applied egg-rr 46.0%

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

      1. unpow-prod-down 44.9%

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

      2. pow2 44.9%

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

      3. add-sqr-sqrt 80.0%

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

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot \left(\ell \cdot \ell\right)}{{k\_m}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + \frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-194)
    (*
     2.0
     (/
      (* (cos k_m) (* l l))
      (* (pow k_m 2.0) (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))
    (/
     2.0
     (*
      (+ 1.0 (+ 1.0 (* (/ k_m t_m) (/ k_m t_m))))
      (* (tan k_m) (* (sin k_m) (pow (/ (pow t_m 1.5) 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 <= 2.6e-194) {
		tmp = 2.0 * ((cos(k_m) * (l * l)) / (pow(k_m, 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * pow((pow(t_m, 1.5) / 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 <= 2.6d-194) then
        tmp = 2.0d0 * ((cos(k_m) * (l * l)) / ((k_m ** 2.0d0) * (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * (((t_m ** 1.5d0) / 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 <= 2.6e-194) {
		tmp = 2.0 * ((Math.cos(k_m) * (l * l)) / (Math.pow(k_m, 2.0) * (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (Math.tan(k_m) * (Math.sin(k_m) * Math.pow((Math.pow(t_m, 1.5) / 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 <= 2.6e-194:
		tmp = 2.0 * ((math.cos(k_m) * (l * l)) / (math.pow(k_m, 2.0) * (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	else:
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (math.tan(k_m) * (math.sin(k_m) * math.pow((math.pow(t_m, 1.5) / 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 <= 2.6e-194)
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(l * l)) / Float64((k_m ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) * Float64(k_m / t_m)))) * Float64(tan(k_m) * Float64(sin(k_m) * (Float64((t_m ^ 1.5) / 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 <= 2.6e-194)
		tmp = 2.0 * ((cos(k_m) * (l * l)) / ((k_m ^ 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t_m) * (k_m / t_m)))) * (tan(k_m) * (sin(k_m) * (((t_m ^ 1.5) / 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, 2.6e-194], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.60000000000000002e-194

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0 65.8%

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

      1. unpow2 65.8%

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

      2. sin-mult 60.2%

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

   5. Applied egg-rr 60.2%

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

      1. div-sub 60.2%

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

      2. +-inverses 60.2%

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

      3. cos-0 60.2%

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

      4. metadata-eval 60.2%

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

      5. count-2 60.2%

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

   7. Simplified 60.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 60.2%

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

   9. Applied egg-rr 60.2%

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

   if 2.60000000000000002e-194 < t

   1. Initial program 63.4%

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

      1. add-sqr-sqrt 35.0%

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

      2. pow2 35.0%

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

      3. associate-/r* 41.7%

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

      4. *-commutative 41.7%

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

      5. sqrt-prod 41.7%

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

      6. associate-/r* 35.0%

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

      7. sqrt-div 35.0%

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

      8. sqrt-pow1 38.6%

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

      9. metadata-eval 38.6%

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

      10. sqrt-prod 27.3%

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

      11. add-sqr-sqrt 46.6%

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

   4. Applied egg-rr 46.6%

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

      1. *-commutative 46.6%

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

   6. Simplified 46.6%

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

      1. unpow2 46.6%

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

   8. Applied egg-rr 46.6%

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

      1. unpow-prod-down 45.7%

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

      2. pow2 45.7%

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

      3. add-sqr-sqrt 79.6%

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

   10. Applied egg-rr 79.6%

       \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k\_m}}{\sin k\_m \cdot {t\_m}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-55)
    (*
     2.0
     (/ (* (cos k_m) (pow l 2.0)) (* (* k_m k_m) (* t_m (pow (sin k_m) 2.0)))))
    (*
     (* l (/ (/ 2.0 (tan k_m)) (* (sin k_m) (pow t_m 3.0))))
     (/ l (+ 2.0 (pow (/ k_m t_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.6e-55) {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / ((k_m * k_m) * (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = (l * ((2.0 / tan(k_m)) / (sin(k_m) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.6d-55) then
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / ((k_m * k_m) * (t_m * (sin(k_m) ** 2.0d0))))
    else
        tmp = (l * ((2.0d0 / tan(k_m)) / (sin(k_m) * (t_m ** 3.0d0)))) * (l / (2.0d0 + ((k_m / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.6e-55) {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = (l * ((2.0 / Math.tan(k_m)) / (Math.sin(k_m) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.6e-55:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * math.pow(math.sin(k_m), 2.0))))
	else:
		tmp = (l * ((2.0 / math.tan(k_m)) / (math.sin(k_m) * math.pow(t_m, 3.0)))) * (l / (2.0 + math.pow((k_m / t_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.6e-55)
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(Float64(k_m * k_m) * Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / tan(k_m)) / Float64(sin(k_m) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.6e-55)
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / ((k_m * k_m) * (t_m * (sin(k_m) ^ 2.0))));
	else
		tmp = (l * ((2.0 / tan(k_m)) / (sin(k_m) * (t_m ^ 3.0)))) * (l / (2.0 + ((k_m / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-55], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5999999999999999e-55

   1. Initial program 54.9%

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

   3. Taylor expanded in t around 0 66.4%

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

      1. unpow2 66.4%

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

   5. Applied egg-rr 66.4%

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

   if 2.5999999999999999e-55 < t

   1. Initial program 69.6%

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

   3. Simplified 67.4%

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

      1. associate-*r* 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 78.0%

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

      4. div-inv 78.0%

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

      5. frac-times 77.9%

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

      6. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. /-rgt-identity 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-/r* 78.0%

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

      4. *-commutative 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.6 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k\_m \cdot \left(\sin k\_m \cdot {t\_m}^{3}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.6e-55)
    (*
     2.0
     (/ (* (cos k_m) (pow l 2.0)) (* (* k_m k_m) (* t_m (pow (sin k_m) 2.0)))))
    (*
     (/ l (+ 2.0 (pow (/ k_m t_m) 2.0)))
     (* l (/ 2.0 (* (tan k_m) (* (sin k_m) (pow t_m 3.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 9.6e-55) {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / ((k_m * k_m) * (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = (l / (2.0 + pow((k_m / t_m), 2.0))) * (l * (2.0 / (tan(k_m) * (sin(k_m) * pow(t_m, 3.0)))));
	}
	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 <= 9.6d-55) then
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / ((k_m * k_m) * (t_m * (sin(k_m) ** 2.0d0))))
    else
        tmp = (l / (2.0d0 + ((k_m / t_m) ** 2.0d0))) * (l * (2.0d0 / (tan(k_m) * (sin(k_m) * (t_m ** 3.0d0)))))
    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 <= 9.6e-55) {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = (l / (2.0 + Math.pow((k_m / t_m), 2.0))) * (l * (2.0 / (Math.tan(k_m) * (Math.sin(k_m) * Math.pow(t_m, 3.0)))));
	}
	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 <= 9.6e-55:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * math.pow(math.sin(k_m), 2.0))))
	else:
		tmp = (l / (2.0 + math.pow((k_m / t_m), 2.0))) * (l * (2.0 / (math.tan(k_m) * (math.sin(k_m) * math.pow(t_m, 3.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 9.6e-55)
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(Float64(k_m * k_m) * Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k_m) * Float64(sin(k_m) * (t_m ^ 3.0))))));
	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 <= 9.6e-55)
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / ((k_m * k_m) * (t_m * (sin(k_m) ^ 2.0))));
	else
		tmp = (l / (2.0 + ((k_m / t_m) ^ 2.0))) * (l * (2.0 / (tan(k_m) * (sin(k_m) * (t_m ^ 3.0)))));
	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, 9.6e-55], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.59999999999999966e-55

   1. Initial program 54.9%

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

   3. Taylor expanded in t around 0 66.4%

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

      1. unpow2 66.4%

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

   5. Applied egg-rr 66.4%

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

   if 9.59999999999999966e-55 < t

   1. Initial program 69.6%

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

   3. Simplified 67.4%

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

      1. associate-*r* 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 78.0%

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

      4. div-inv 78.0%

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

      5. frac-times 77.9%

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

      6. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. /-rgt-identity 77.9%

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

      2. *-commutative 77.9%

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

      3. *-commutative 77.9%

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

      4. *-commutative 77.9%

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

   8. Simplified 77.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left({\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(k\_m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.45e+20)
    (/ 2.0 (* (sin k_m) (* (pow (/ (pow t_m 1.5) l) 2.0) (* k_m 2.0))))
    (*
     2.0
     (/
      (* (cos k_m) (pow l 2.0))
      (* (* k_m k_m) (* t_m (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+20) {
		tmp = 2.0 / (sin(k_m) * (pow((pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / ((k_m * k_m) * (t_m * pow(sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d+20) then
        tmp = 2.0d0 / (sin(k_m) * ((((t_m ** 1.5d0) / l) ** 2.0d0) * (k_m * 2.0d0)))
    else
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / ((k_m * k_m) * (t_m * (sin(k_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+20) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * Math.pow(Math.sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.45e+20:
		tmp = 2.0 / (math.sin(k_m) * (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)))
	else:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / ((k_m * k_m) * (t_m * math.pow(math.sin(k_m), 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e+20)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(k_m * 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(Float64(k_m * k_m) * Float64(t_m * (sin(k_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e+20)
		tmp = 2.0 / (sin(k_m) * ((((t_m ^ 1.5) / l) ^ 2.0) * (k_m * 2.0)));
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / ((k_m * k_m) * (t_m * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.45e+20], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e20

   1. Initial program 61.8%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. associate-/r* 37.1%

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

      4. *-commutative 37.1%

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

      5. sqrt-prod 20.0%

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

      6. associate-/r* 16.8%

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

      7. sqrt-div 16.8%

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

      8. sqrt-pow1 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-prod 13.0%

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

      11. add-sqr-sqrt 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. *-commutative 22.2%

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

   6. Simplified 22.2%

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

      1. div-inv 22.2%

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

      2. associate-*l* 22.2%

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

      3. *-commutative 22.2%

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

      4. unpow-prod-down 21.7%

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

      5. pow2 21.7%

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

      6. add-sqr-sqrt 39.0%

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

      7. add-sqr-sqrt 39.0%

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

      8. pow2 39.0%

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

   8. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 39.0%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 38.6%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 38.6%

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

   11. Taylor expanded in k around 0 36.3%

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

   if 2.45e20 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

   5. Applied egg-rr 75.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(k \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left({\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(k\_m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot \left(\ell \cdot \ell\right)}{{k\_m}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.45e+20)
    (/ 2.0 (* (sin k_m) (* (pow (/ (pow t_m 1.5) l) 2.0) (* k_m 2.0))))
    (*
     2.0
     (/
      (* (cos k_m) (* l l))
      (* (pow k_m 2.0) (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+20) {
		tmp = 2.0 / (sin(k_m) * (pow((pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((cos(k_m) * (l * l)) / (pow(k_m, 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d+20) then
        tmp = 2.0d0 / (sin(k_m) * ((((t_m ** 1.5d0) / l) ** 2.0d0) * (k_m * 2.0d0)))
    else
        tmp = 2.0d0 * ((cos(k_m) * (l * l)) / ((k_m ** 2.0d0) * (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+20) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * (l * l)) / (Math.pow(k_m, 2.0) * (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.45e+20:
		tmp = 2.0 / (math.sin(k_m) * (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)))
	else:
		tmp = 2.0 * ((math.cos(k_m) * (l * l)) / (math.pow(k_m, 2.0) * (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e+20)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(k_m * 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(l * l)) / Float64((k_m ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e+20)
		tmp = 2.0 / (sin(k_m) * ((((t_m ^ 1.5) / l) ^ 2.0) * (k_m * 2.0)));
	else
		tmp = 2.0 * ((cos(k_m) * (l * l)) / ((k_m ^ 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.45e+20], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e20

   1. Initial program 61.8%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. associate-/r* 37.1%

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

      4. *-commutative 37.1%

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

      5. sqrt-prod 20.0%

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

      6. associate-/r* 16.8%

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

      7. sqrt-div 16.8%

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

      8. sqrt-pow1 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-prod 13.0%

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

      11. add-sqr-sqrt 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. *-commutative 22.2%

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

   6. Simplified 22.2%

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

      1. div-inv 22.2%

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

      2. associate-*l* 22.2%

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

      3. *-commutative 22.2%

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

      4. unpow-prod-down 21.7%

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

      5. pow2 21.7%

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

      6. add-sqr-sqrt 39.0%

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

      7. add-sqr-sqrt 39.0%

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

      8. pow2 39.0%

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

   8. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 39.0%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 38.6%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 38.6%

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

   11. Taylor expanded in k around 0 36.3%

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

   if 2.45e20 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

      2. sin-mult 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. div-sub 75.5%

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

      2. +-inverses 75.5%

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

      3. cos-0 75.5%

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

      4. metadata-eval 75.5%

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

      5. count-2 75.5%

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

   7. Simplified 75.5%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 75.5%

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

   9. Applied egg-rr 75.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(k \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left({\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(k\_m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k\_m}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e+99)
    (/ 2.0 (* (sin k_m) (* (pow (/ (pow t_m 1.5) l) 2.0) (* k_m 2.0))))
    (*
     2.0
     (/
      (pow l 2.0)
      (* (pow k_m 2.0) (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+99) {
		tmp = 2.0 / (sin(k_m) * (pow((pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k_m, 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d+99) then
        tmp = 2.0d0 / (sin(k_m) * ((((t_m ** 1.5d0) / l) ** 2.0d0) * (k_m * 2.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k_m ** 2.0d0) * (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+99) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k_m, 2.0) * (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e+99:
		tmp = 2.0 / (math.sin(k_m) * (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k_m, 2.0) * (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e+99)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(k_m * 2.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k_m ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e+99)
		tmp = 2.0 / (sin(k_m) * ((((t_m ^ 1.5) / l) ^ 2.0) * (k_m * 2.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) / ((k_m ^ 2.0) * (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e+99], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000004e99

   1. Initial program 62.3%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. associate-/r* 37.2%

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

      4. *-commutative 37.2%

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

      5. sqrt-prod 20.2%

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

      6. associate-/r* 17.2%

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

      7. sqrt-div 17.2%

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

      8. sqrt-pow1 18.6%

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

      9. metadata-eval 18.6%

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

      10. sqrt-prod 13.2%

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

      11. add-sqr-sqrt 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. *-commutative 22.3%

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

   6. Simplified 22.3%

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

      1. div-inv 22.3%

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

      2. associate-*l* 22.3%

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

      3. *-commutative 22.3%

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

      4. unpow-prod-down 21.8%

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

      5. pow2 21.8%

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

      6. add-sqr-sqrt 39.4%

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

      7. add-sqr-sqrt 39.4%

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

      8. pow2 39.4%

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

   8. Applied egg-rr 39.4%

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

      1. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 39.4%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 39.1%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 39.1%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 39.1%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 39.1%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 39.1%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 39.1%

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

   11. Taylor expanded in k around 0 37.0%

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

   if 2.50000000000000004e99 < k

   1. Initial program 48.1%

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

   3. Taylor expanded in t around 0 72.8%

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

      1. unpow2 72.8%

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

      2. sin-mult 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. div-sub 72.6%

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

      2. +-inverses 72.6%

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

      3. cos-0 72.6%

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

      4. metadata-eval 72.6%

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

      5. count-2 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in k around 0 59.5%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(k \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k\_m}}{k\_m \cdot {t\_m}^{3}}}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.6e-162)
    (/
     (* (* l l) (/ (/ 2.0 k_m) (* k_m (pow t_m 3.0))))
     (+ 2.0 (pow (/ k_m t_m) 2.0)))
    (if (<= k_m 2.9e+20)
      (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k_m k_m))))
      (* 2.0 (/ (* (cos k_m) (pow l 2.0)) (* t_m (pow k_m 4.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-162) {
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * pow(t_m, 3.0)))) / (2.0 + pow((k_m / t_m), 2.0));
	} else if (k_m <= 2.9e+20) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	} else {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.6d-162) then
        tmp = ((l * l) * ((2.0d0 / k_m) / (k_m * (t_m ** 3.0d0)))) / (2.0d0 + ((k_m / t_m) ** 2.0d0))
    else if (k_m <= 2.9d+20) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k_m * k_m)))
    else
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-162) {
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k_m / t_m), 2.0));
	} else if (k_m <= 2.9e+20) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.6e-162:
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * math.pow(t_m, 3.0)))) / (2.0 + math.pow((k_m / t_m), 2.0))
	elif k_m <= 2.9e+20:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)))
	else:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.6e-162)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k_m) / Float64(k_m * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)));
	elseif (k_m <= 2.9e+20)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.6e-162)
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * (t_m ^ 3.0)))) / (2.0 + ((k_m / t_m) ^ 2.0));
	elseif (k_m <= 2.9e+20)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k_m * k_m)));
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.6e-162], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.9e+20], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.6e-162

   1. Initial program 60.4%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in k around 0 56.7%

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

   6. Taylor expanded in k around 0 56.7%

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

   if 2.6e-162 < k < 2.9e20

   1. Initial program 68.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in k around 0 81.8%

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

      1. unpow2 58.5%

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

   7. Applied egg-rr 81.8%

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

      1. add-sqr-sqrt 38.1%

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

      2. pow2 38.1%

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

      3. associate-/r* 33.0%

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

      4. sqrt-div 30.3%

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

      5. sqrt-pow1 30.4%

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

      6. metadata-eval 30.4%

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

      7. sqrt-prod 24.6%

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

      8. add-sqr-sqrt 35.5%

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

   9. Applied egg-rr 35.5%

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

   if 2.9e20 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 75.6%

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

   4. Taylor expanded in k around 0 60.6%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left({\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(k\_m \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.8e+20)
    (/ 2.0 (* (sin k_m) (* (pow (/ (pow t_m 1.5) l) 2.0) (* k_m 2.0))))
    (* 2.0 (/ (* (cos k_m) (pow l 2.0)) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e+20) {
		tmp = 2.0 / (sin(k_m) * (pow((pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.8d+20) then
        tmp = 2.0d0 / (sin(k_m) * ((((t_m ** 1.5d0) / l) ** 2.0d0) * (k_m * 2.0d0)))
    else
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e+20) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.8e+20:
		tmp = 2.0 / (math.sin(k_m) * (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (k_m * 2.0)))
	else:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.8e+20)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(k_m * 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.8e+20)
		tmp = 2.0 / (sin(k_m) * ((((t_m ^ 1.5) / l) ^ 2.0) * (k_m * 2.0)));
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.8e+20], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.8e20

   1. Initial program 61.8%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. associate-/r* 37.1%

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

      4. *-commutative 37.1%

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

      5. sqrt-prod 20.0%

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

      6. associate-/r* 16.8%

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

      7. sqrt-div 16.8%

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

      8. sqrt-pow1 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-prod 13.0%

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

      11. add-sqr-sqrt 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. *-commutative 22.2%

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

   6. Simplified 22.2%

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

      1. div-inv 22.2%

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

      2. associate-*l* 22.2%

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

      3. *-commutative 22.2%

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

      4. unpow-prod-down 21.7%

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

      5. pow2 21.7%

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

      6. add-sqr-sqrt 39.0%

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

      7. add-sqr-sqrt 39.0%

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

      8. pow2 39.0%

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

   8. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]

      2. metadata-eval 39.0%

         \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} \]

      3. associate-*l* 38.6%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      4. unpow2 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}\right)\right)} \]

      5. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)\right)} \]

      6. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)} \cdot \color{blue}{\sqrt{1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)}}\right)\right)\right)} \]

      7. rem-square-sqrt 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \color{blue}{\left(1 \cdot 1 + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\right)} \]

      8. metadata-eval 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(\color{blue}{1} + \mathsf{hypot}\left(1, \frac{k}{t}\right) \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      9. hypot-undefine 38.6%

         \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\sqrt{1 \cdot 1 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      10. metadata-eval 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{1} + \frac{k}{t} \cdot \frac{k}{t}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      11. unpow2 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

      12. +-commutative 38.6%

          \[\leadsto \frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \sqrt{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 1}} \cdot \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)} \]

   10. Simplified 38.6%

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

   11. Taylor expanded in k around 0 36.3%

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

   if 2.8e20 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 75.6%

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

   4. Taylor expanded in k around 0 60.6%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(k \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k\_m}}{k\_m \cdot {t\_m}^{3}}}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.1e-162)
    (/
     (* (* l l) (/ (/ 2.0 k_m) (* k_m (pow t_m 3.0))))
     (+ 2.0 (pow (/ k_m t_m) 2.0)))
    (if (<= k_m 5e+69)
      (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k_m k_m))))
      (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e-162) {
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * pow(t_m, 3.0)))) / (2.0 + pow((k_m / t_m), 2.0));
	} else if (k_m <= 5e+69) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	} else {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.1d-162) then
        tmp = ((l * l) * ((2.0d0 / k_m) / (k_m * (t_m ** 3.0d0)))) / (2.0d0 + ((k_m / t_m) ** 2.0d0))
    else if (k_m <= 5d+69) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k_m * k_m)))
    else
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k_m ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e-162) {
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k_m / t_m), 2.0));
	} else if (k_m <= 5e+69) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	} else {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.1e-162:
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * math.pow(t_m, 3.0)))) / (2.0 + math.pow((k_m / t_m), 2.0))
	elif k_m <= 5e+69:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)))
	else:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k_m, 4.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.1e-162)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k_m) / Float64(k_m * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)));
	elseif (k_m <= 5e+69)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k_m ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.1e-162)
		tmp = ((l * l) * ((2.0 / k_m) / (k_m * (t_m ^ 3.0)))) / (2.0 + ((k_m / t_m) ^ 2.0));
	elseif (k_m <= 5e+69)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k_m * k_m)));
	else
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k_m ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.1e-162], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5e+69], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.1e-162

   1. Initial program 60.4%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in k around 0 56.7%

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

   6. Taylor expanded in k around 0 56.7%

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

   if 2.1e-162 < k < 5.00000000000000036e69

   1. Initial program 68.7%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in k around 0 78.0%

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

      1. unpow2 62.9%

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

   7. Applied egg-rr 78.0%

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

      1. add-sqr-sqrt 38.9%

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

      2. pow2 38.9%

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

      3. associate-/r* 34.6%

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

      4. sqrt-div 32.3%

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

      5. sqrt-pow1 32.3%

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

      6. metadata-eval 32.3%

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

      7. sqrt-prod 25.2%

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

      8. add-sqr-sqrt 36.7%

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

   9. Applied egg-rr 36.7%

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

   if 5.00000000000000036e69 < k

   1. Initial program 50.6%

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

   3. Taylor expanded in t around 0 74.2%

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

      1. unpow2 74.2%

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

      2. sin-mult 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. div-sub 74.0%

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

      2. +-inverses 74.0%

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

      3. cos-0 74.0%

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

      4. metadata-eval 74.0%

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

      5. count-2 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in k around 0 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. times-frac 58.8%

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

   10. Simplified 58.8%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-80)
    (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k_m 4.0)))
    (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k_m k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.35e-80) {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k_m, 4.0));
	} else {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.35d-80) then
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k_m ** 4.0d0))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k_m * k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.35e-80) {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k_m, 4.0));
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.35e-80:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k_m, 4.0))
	else:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k_m * k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.35e-80)
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k_m ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.35e-80)
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k_m ^ 4.0));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k_m * k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-80], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.34999999999999986e-80

   1. Initial program 53.8%

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

   3. Taylor expanded in t around 0 65.5%

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

      1. unpow2 65.5%

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

      2. sin-mult 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. div-sub 60.5%

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

      2. +-inverses 60.5%

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

      3. cos-0 60.5%

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

      4. metadata-eval 60.5%

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

      5. count-2 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in k around 0 56.3%

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

      1. associate-*r/ 55.7%

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

      2. *-commutative 55.7%

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

      3. times-frac 57.4%

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

   10. Simplified 57.4%

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

   if 2.34999999999999986e-80 < t

   1. Initial program 71.0%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in k around 0 62.2%

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

      1. unpow2 52.7%

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

   7. Applied egg-rr 62.2%

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

      1. add-sqr-sqrt 62.2%

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

      2. pow2 62.2%

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

      3. associate-/r* 56.5%

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

      4. sqrt-div 56.4%

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

      5. sqrt-pow1 56.5%

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

      6. metadata-eval 56.5%

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

      7. sqrt-prod 31.0%

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

      8. add-sqr-sqrt 64.3%

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

   9. Applied egg-rr 64.3%

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

Alternative 17: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-83)
    (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k_m 4.0)))
    (/ 2.0 (* (* 2.0 (* k_m k_m)) (* (/ t_m l) (/ (* t_m t_m) l)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.8e-83) {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k_m, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.8d-83) then
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k_m ** 4.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.8e-83) {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k_m, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.8e-83:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k_m, 4.0))
	else:
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.8e-83)
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k_m ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.8e-83)
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k_m ^ 4.0));
	else
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-83], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-83

   1. Initial program 53.8%

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

   3. Taylor expanded in t around 0 65.5%

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

      1. unpow2 65.5%

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

      2. sin-mult 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. div-sub 60.5%

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

      2. +-inverses 60.5%

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

      3. cos-0 60.5%

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

      4. metadata-eval 60.5%

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

      5. count-2 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in k around 0 56.3%

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

      1. associate-*r/ 55.7%

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

      2. *-commutative 55.7%

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

      3. times-frac 57.4%

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

   10. Simplified 57.4%

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

   if 2.8000000000000001e-83 < t

   1. Initial program 71.0%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in k around 0 62.2%

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

      1. unpow2 52.7%

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

   7. Applied egg-rr 62.2%

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

      1. associate-/r* 56.4%

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

      2. unpow3 56.4%

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

      3. times-frac 62.2%

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

      4. pow2 62.2%

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

   9. Applied egg-rr 62.2%

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

       1. unpow2 62.2%

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

   11. Applied egg-rr 62.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.1e+58)
    (/ 2.0 (* (* 2.0 (* k_m k_m)) (* (/ t_m l) (/ (* t_m t_m) l))))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e+58) {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.1d+58) then
        tmp = 2.0d0 / ((2.0d0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e+58) {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.1e+58:
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.1e+58)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.1e+58)
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.1e+58], N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.10000000000000012e58

   1. Initial program 61.8%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in k around 0 59.5%

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

      1. unpow2 57.4%

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

   7. Applied egg-rr 59.5%

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

      1. associate-/r* 54.3%

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

      2. unpow3 54.3%

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

      3. times-frac 61.5%

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

      4. pow2 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. unpow2 61.5%

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

   11. Applied egg-rr 61.5%

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

   if 2.10000000000000012e58 < k

   1. Initial program 52.5%

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

   3. Taylor expanded in t around 0 75.1%

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

   4. Taylor expanded in k around 0 58.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.1% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

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 / ((2.0d0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))))
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 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))));
}

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 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))))

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(2.0 * Float64(k_m * k_m)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l)))))
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 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))));
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[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Simplified 59.7%

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

5. Taylor expanded in k around 0 57.4%

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

   1. unpow2 61.0%

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

7. Applied egg-rr 57.4%

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

   1. associate-/r* 52.8%

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

   2. unpow3 52.8%

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

   3. times-frac 59.0%

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

   4. pow2 59.0%

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

9. Applied egg-rr 59.0%

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

    1. unpow2 59.0%

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

11. Applied egg-rr 59.0%

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

12. Final simplification 59.0%

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

Reproduce

herbie shell --seed 2024156 
(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))))