Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 88.2%

Time: 20.8s

Alternatives: 19

Speedup: 3.8×

• 39.2% 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: 36.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{2}{t\_m}}}{k\_m \cdot \frac{\sin k\_m}{\ell \cdot \sqrt{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{{\sin k\_m}^{2}}{\cos k\_m \cdot {\ell}^{2}} \cdot {k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 1.9e-14)
    (pow
     (/ (sqrt (/ 2.0 t_m)) (* k_m (/ (sin k_m) (* l (sqrt (cos k_m))))))
     2.0)
    (if (<= k_m 2.5e+152)
      (/
       2.0
       (*
        t_m
        (* (/ (pow (sin k_m) 2.0) (* (cos k_m) (pow l 2.0))) (pow k_m 2.0))))
      (/
       2.0
       (*
        (pow (* (/ k_m t_m) (/ (pow t_m 1.5) l)) 2.0)
        (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = pow((sqrt((2.0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))), 2.0);
	} else if (k_m <= 2.5e+152) {
		tmp = 2.0 / (t_m * ((pow(sin(k_m), 2.0) / (cos(k_m) * pow(l, 2.0))) * pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / (pow(((k_m / t_m) * (pow(t_m, 1.5) / l)), 2.0) * (sin(k_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-14) then
        tmp = (sqrt((2.0d0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))) ** 2.0d0
    else if (k_m <= 2.5d+152) then
        tmp = 2.0d0 / (t_m * (((sin(k_m) ** 2.0d0) / (cos(k_m) * (l ** 2.0d0))) * (k_m ** 2.0d0)))
    else
        tmp = 2.0d0 / ((((k_m / t_m) * ((t_m ** 1.5d0) / l)) ** 2.0d0) * (sin(k_m) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = Math.pow((Math.sqrt((2.0 / t_m)) / (k_m * (Math.sin(k_m) / (l * Math.sqrt(Math.cos(k_m)))))), 2.0);
	} else if (k_m <= 2.5e+152) {
		tmp = 2.0 / (t_m * ((Math.pow(Math.sin(k_m), 2.0) / (Math.cos(k_m) * Math.pow(l, 2.0))) * Math.pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((k_m / t_m) * (Math.pow(t_m, 1.5) / l)), 2.0) * (Math.sin(k_m) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.9e-14:
		tmp = math.pow((math.sqrt((2.0 / t_m)) / (k_m * (math.sin(k_m) / (l * math.sqrt(math.cos(k_m)))))), 2.0)
	elif k_m <= 2.5e+152:
		tmp = 2.0 / (t_m * ((math.pow(math.sin(k_m), 2.0) / (math.cos(k_m) * math.pow(l, 2.0))) * math.pow(k_m, 2.0)))
	else:
		tmp = 2.0 / (math.pow(((k_m / t_m) * (math.pow(t_m, 1.5) / l)), 2.0) * (math.sin(k_m) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-14)
		tmp = Float64(sqrt(Float64(2.0 / t_m)) / Float64(k_m * Float64(sin(k_m) / Float64(l * sqrt(cos(k_m)))))) ^ 2.0;
	elseif (k_m <= 2.5e+152)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64((sin(k_m) ^ 2.0) / Float64(cos(k_m) * (l ^ 2.0))) * (k_m ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(k_m / t_m) * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(sin(k_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.9e-14)
		tmp = (sqrt((2.0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))) ^ 2.0;
	elseif (k_m <= 2.5e+152)
		tmp = 2.0 / (t_m * (((sin(k_m) ^ 2.0) / (cos(k_m) * (l ^ 2.0))) * (k_m ^ 2.0)));
	else
		tmp = 2.0 / ((((k_m / t_m) * ((t_m ^ 1.5) / l)) ^ 2.0) * (sin(k_m) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-14], N[Power[N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 2.5e+152], N[(2.0 / N[(t$95$m * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9000000000000001e-14

   1. Initial program 41.4%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

   4. Applied egg-rr 46.8%

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

      1. add-exp-log 46.5%

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

      2. expm1-define 46.5%

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

      3. log1p-define 51.3%

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

      4. unpow2 51.3%

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

      5. expm1-log1p-u 51.6%

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

      6. clear-num 51.6%

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

      7. un-div-inv 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in t around 0 76.7%

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

      1. associate-/l* 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-*r/ 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-*l* 77.3%

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

      6. *-commutative 77.3%

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

   9. Simplified 77.3%

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

       1. add-sqr-sqrt 48.7%

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

       2. pow2 48.7%

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

   11. Applied egg-rr 38.5%

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

   if 1.9000000000000001e-14 < k < 2.5e152

   1. Initial program 16.2%

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

      1. associate-/r* 22.2%

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

      2. div-inv 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. add-exp-log 21.6%

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

      2. expm1-define 21.6%

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

      3. log1p-define 31.8%

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

      4. unpow2 31.8%

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

      5. expm1-log1p-u 32.5%

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

      6. clear-num 32.5%

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

      7. un-div-inv 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in t around 0 78.3%

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

      1. associate-/l* 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-*l* 81.9%

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

      6. *-commutative 81.9%

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

   9. Simplified 81.9%

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

   if 2.5e152 < k

   1. Initial program 40.5%

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

      1. add-sqr-sqrt 17.3%

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

      2. pow2 17.3%

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

   4. Applied egg-rr 14.2%

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

      1. associate-*r* 14.2%

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

      2. unpow-prod-down 14.2%

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

      3. pow2 14.2%

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

      4. add-sqr-sqrt 34.3%

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

   6. Applied egg-rr 34.3%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{2}{t}}}{k \cdot \frac{\sin k}{\ell \cdot \sqrt{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\cos k \cdot {\ell}^{2}} \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{2}{t\_m}}}{k\_m \cdot \frac{\sin k\_m}{\ell \cdot \sqrt{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 4.05 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{{\left(k\_m \cdot \sin k\_m\right)}^{2}}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 1.9e-14)
    (pow
     (/ (sqrt (/ 2.0 t_m)) (* k_m (/ (sin k_m) (* l (sqrt (cos k_m))))))
     2.0)
    (if (<= k_m 4.05e+152)
      (/ 2.0 (* t_m (/ (pow (* k_m (sin k_m)) 2.0) (* (cos k_m) (pow l 2.0)))))
      (/
       2.0
       (*
        (pow (* (/ k_m t_m) (/ (pow t_m 1.5) l)) 2.0)
        (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = pow((sqrt((2.0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))), 2.0);
	} else if (k_m <= 4.05e+152) {
		tmp = 2.0 / (t_m * (pow((k_m * sin(k_m)), 2.0) / (cos(k_m) * pow(l, 2.0))));
	} else {
		tmp = 2.0 / (pow(((k_m / t_m) * (pow(t_m, 1.5) / l)), 2.0) * (sin(k_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-14) then
        tmp = (sqrt((2.0d0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))) ** 2.0d0
    else if (k_m <= 4.05d+152) then
        tmp = 2.0d0 / (t_m * (((k_m * sin(k_m)) ** 2.0d0) / (cos(k_m) * (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((k_m / t_m) * ((t_m ** 1.5d0) / l)) ** 2.0d0) * (sin(k_m) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = Math.pow((Math.sqrt((2.0 / t_m)) / (k_m * (Math.sin(k_m) / (l * Math.sqrt(Math.cos(k_m)))))), 2.0);
	} else if (k_m <= 4.05e+152) {
		tmp = 2.0 / (t_m * (Math.pow((k_m * Math.sin(k_m)), 2.0) / (Math.cos(k_m) * Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / (Math.pow(((k_m / t_m) * (Math.pow(t_m, 1.5) / l)), 2.0) * (Math.sin(k_m) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.9e-14:
		tmp = math.pow((math.sqrt((2.0 / t_m)) / (k_m * (math.sin(k_m) / (l * math.sqrt(math.cos(k_m)))))), 2.0)
	elif k_m <= 4.05e+152:
		tmp = 2.0 / (t_m * (math.pow((k_m * math.sin(k_m)), 2.0) / (math.cos(k_m) * math.pow(l, 2.0))))
	else:
		tmp = 2.0 / (math.pow(((k_m / t_m) * (math.pow(t_m, 1.5) / l)), 2.0) * (math.sin(k_m) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-14)
		tmp = Float64(sqrt(Float64(2.0 / t_m)) / Float64(k_m * Float64(sin(k_m) / Float64(l * sqrt(cos(k_m)))))) ^ 2.0;
	elseif (k_m <= 4.05e+152)
		tmp = Float64(2.0 / Float64(t_m * Float64((Float64(k_m * sin(k_m)) ^ 2.0) / Float64(cos(k_m) * (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(k_m / t_m) * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(sin(k_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.9e-14)
		tmp = (sqrt((2.0 / t_m)) / (k_m * (sin(k_m) / (l * sqrt(cos(k_m)))))) ^ 2.0;
	elseif (k_m <= 4.05e+152)
		tmp = 2.0 / (t_m * (((k_m * sin(k_m)) ^ 2.0) / (cos(k_m) * (l ^ 2.0))));
	else
		tmp = 2.0 / ((((k_m / t_m) * ((t_m ^ 1.5) / l)) ^ 2.0) * (sin(k_m) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-14], N[Power[N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 4.05e+152], N[(2.0 / N[(t$95$m * N[(N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9000000000000001e-14

   1. Initial program 41.4%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

   4. Applied egg-rr 46.8%

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

      1. add-exp-log 46.5%

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

      2. expm1-define 46.5%

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

      3. log1p-define 51.3%

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

      4. unpow2 51.3%

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

      5. expm1-log1p-u 51.6%

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

      6. clear-num 51.6%

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

      7. un-div-inv 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in t around 0 76.7%

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

      1. associate-/l* 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-*r/ 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-*l* 77.3%

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

      6. *-commutative 77.3%

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

   9. Simplified 77.3%

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

       1. add-sqr-sqrt 48.7%

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

       2. pow2 48.7%

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

   11. Applied egg-rr 38.5%

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

   if 1.9000000000000001e-14 < k < 4.04999999999999999e152

   1. Initial program 16.2%

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

      1. associate-/r* 22.2%

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

      2. div-inv 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. add-exp-log 21.6%

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

      2. expm1-define 21.6%

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

      3. log1p-define 31.8%

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

      4. unpow2 31.8%

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

      5. expm1-log1p-u 32.5%

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

      6. clear-num 32.5%

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

      7. un-div-inv 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in t around 0 78.3%

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

      1. associate-/l* 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-*l* 81.9%

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

      6. *-commutative 81.9%

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

   9. Simplified 81.9%

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

       1. associate-*l/ 81.9%

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

       2. pow-prod-down 81.8%

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

   11. Applied egg-rr 81.8%

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

   if 4.04999999999999999e152 < k

   1. Initial program 40.5%

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

      1. add-sqr-sqrt 17.3%

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

      2. pow2 17.3%

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

   4. Applied egg-rr 14.2%

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

      1. associate-*r* 14.2%

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

      2. unpow-prod-down 14.2%

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

      3. pow2 14.2%

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

      4. add-sqr-sqrt 34.3%

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

   6. Applied egg-rr 34.3%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{2}{t}}}{k \cdot \frac{\sin k}{\ell \cdot \sqrt{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 4.05 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-219)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+300)
      (/
       2.0
       (*
        (* k_m k_m)
        (/ (* t_m (pow (sin k_m) 2.0)) (* (cos k_m) (pow l 2.0)))))
      (/
       2.0
       (pow (* (sqrt (/ t_m (cos k_m))) (/ (* k_m (sin k_m)) l)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * pow(sin(k_m), 2.0)) / (cos(k_m) * pow(l, 2.0))));
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 2d-219) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+300) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t_m * (sin(k_m) ** 2.0d0)) / (cos(k_m) * (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / (Math.cos(k_m) * Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * ((k_m * Math.sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 2e-219:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+300:
		tmp = 2.0 / ((k_m * k_m) * ((t_m * math.pow(math.sin(k_m), 2.0)) / (math.cos(k_m) * math.pow(l, 2.0))))
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k_m))) * ((k_m * math.sin(k_m)) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-219)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+300)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / Float64(cos(k_m) * (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(Float64(k_m * sin(k_m)) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 2e-219)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+300)
		tmp = 2.0 / ((k_m * k_m) * ((t_m * (sin(k_m) ^ 2.0)) / (cos(k_m) * (l ^ 2.0))));
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-219], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+300], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.0000000000000001e-219

   1. Initial program 28.8%

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

      1. add-sqr-sqrt 11.0%

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

      2. pow2 11.0%

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

   4. Applied egg-rr 13.9%

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

   5. Taylor expanded in k around 0 38.2%

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

   if 2.0000000000000001e-219 < (*.f64 l l) < 2.0000000000000001e300

   1. Initial program 42.8%

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

   3. Taylor expanded in t around 0 87.5%

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

      1. associate-/l* 88.6%

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

      2. *-commutative 88.6%

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

   5. Simplified 88.6%

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

      1. unpow2 88.6%

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

   7. Applied egg-rr 88.6%

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

   if 2.0000000000000001e300 < (*.f64 l l)

   1. Initial program 40.2%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in k around inf 56.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k\_m}^{2} \cdot \left(t\_m \cdot {k\_m}^{2}\right)}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-219)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+300)
      (*
       (* l l)
       (/ 2.0 (/ (* (pow (sin k_m) 2.0) (* t_m (pow k_m 2.0))) (cos k_m))))
      (/
       2.0
       (pow (* (sqrt (/ t_m (cos k_m))) (/ (* k_m (sin k_m)) l)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = (l * l) * (2.0 / ((pow(sin(k_m), 2.0) * (t_m * pow(k_m, 2.0))) / cos(k_m)));
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 2d-219) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+300) then
        tmp = (l * l) * (2.0d0 / (((sin(k_m) ** 2.0d0) * (t_m * (k_m ** 2.0d0))) / cos(k_m)))
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k_m), 2.0) * (t_m * Math.pow(k_m, 2.0))) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * ((k_m * Math.sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 2e-219:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+300:
		tmp = (l * l) * (2.0 / ((math.pow(math.sin(k_m), 2.0) * (t_m * math.pow(k_m, 2.0))) / math.cos(k_m)))
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k_m))) * ((k_m * math.sin(k_m)) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-219)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+300)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * Float64(t_m * (k_m ^ 2.0))) / cos(k_m))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(Float64(k_m * sin(k_m)) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 2e-219)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+300)
		tmp = (l * l) * (2.0 / (((sin(k_m) ^ 2.0) * (t_m * (k_m ^ 2.0))) / cos(k_m)));
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-219], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+300], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.0000000000000001e-219

   1. Initial program 28.8%

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

      1. add-sqr-sqrt 11.0%

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

      2. pow2 11.0%

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

   4. Applied egg-rr 13.9%

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

   5. Taylor expanded in k around 0 38.2%

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

   if 2.0000000000000001e-219 < (*.f64 l l) < 2.0000000000000001e300

   1. Initial program 42.8%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in t around 0 87.5%

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

      1. associate-*r* 87.5%

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

   7. Simplified 87.5%

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

   if 2.0000000000000001e300 < (*.f64 l l)

   1. Initial program 40.2%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in k around inf 56.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-219)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+300)
      (*
       (* l l)
       (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow (sin k_m) 2.0)) (cos k_m)))))
      (/
       2.0
       (pow (* (sqrt (/ t_m (cos k_m))) (/ (* k_m (sin k_m)) l)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * ((t_m * pow(sin(k_m), 2.0)) / cos(k_m))));
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 2d-219) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+300) then
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * ((t_m * (sin(k_m) ** 2.0d0)) / cos(k_m))))
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-219) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m))));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * ((k_m * Math.sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 2e-219:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+300:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m))))
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k_m))) * ((k_m * math.sin(k_m)) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-219)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+300)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / cos(k_m)))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(Float64(k_m * sin(k_m)) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 2e-219)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+300)
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * ((t_m * (sin(k_m) ^ 2.0)) / cos(k_m))));
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-219], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+300], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.0000000000000001e-219

   1. Initial program 28.8%

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

      1. add-sqr-sqrt 11.0%

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

      2. pow2 11.0%

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

   4. Applied egg-rr 13.9%

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

   5. Taylor expanded in k around 0 38.2%

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

   if 2.0000000000000001e-219 < (*.f64 l l) < 2.0000000000000001e300

   1. Initial program 42.8%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in t around 0 87.5%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   if 2.0000000000000001e300 < (*.f64 l l)

   1. Initial program 40.2%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in k around inf 56.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sin k\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{{t\_2}^{2}}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{t\_2}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sin k_m))))
   (*
    t_s
    (if (<= (* l l) 5e-324)
      (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
      (if (<= (* l l) 2e+300)
        (/ 2.0 (* t_m (/ (pow t_2 2.0) (* (cos k_m) (pow l 2.0)))))
        (/ 2.0 (pow (* (sqrt (/ t_m (cos k_m))) (/ t_2 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 t_2 = k_m * sin(k_m);
	double tmp;
	if ((l * l) <= 5e-324) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = 2.0 / (t_m * (pow(t_2, 2.0) / (cos(k_m) * pow(l, 2.0))));
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k_m))) * (t_2 / 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) :: t_2
    real(8) :: tmp
    t_2 = k_m * sin(k_m)
    if ((l * l) <= 5d-324) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+300) then
        tmp = 2.0d0 / (t_m * ((t_2 ** 2.0d0) / (cos(k_m) * (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k_m))) * (t_2 / 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 t_2 = k_m * Math.sin(k_m);
	double tmp;
	if ((l * l) <= 5e-324) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+300) {
		tmp = 2.0 / (t_m * (Math.pow(t_2, 2.0) / (Math.cos(k_m) * Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * (t_2 / 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):
	t_2 = k_m * math.sin(k_m)
	tmp = 0
	if (l * l) <= 5e-324:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+300:
		tmp = 2.0 / (t_m * (math.pow(t_2, 2.0) / (math.cos(k_m) * math.pow(l, 2.0))))
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k_m))) * (t_2 / 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)
	t_2 = Float64(k_m * sin(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 5e-324)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+300)
		tmp = Float64(2.0 / Float64(t_m * Float64((t_2 ^ 2.0) / Float64(cos(k_m) * (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(t_2 / 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)
	t_2 = k_m * sin(k_m);
	tmp = 0.0;
	if ((l * l) <= 5e-324)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+300)
		tmp = 2.0 / (t_m * ((t_2 ^ 2.0) / (cos(k_m) * (l ^ 2.0))));
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k_m))) * (t_2 / 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_] := Block[{t$95$2 = N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-324], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+300], N[(2.0 / N[(t$95$m * N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.94066e-324

   1. Initial program 25.5%

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

      1. add-sqr-sqrt 9.1%

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

      2. pow2 9.1%

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

   4. Applied egg-rr 13.0%

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

   5. Taylor expanded in k around 0 36.3%

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

   if 4.94066e-324 < (*.f64 l l) < 2.0000000000000001e300

   1. Initial program 42.3%

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

      1. associate-/r* 42.3%

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

      2. div-inv 42.3%

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

   4. Applied egg-rr 42.3%

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

      1. add-exp-log 41.9%

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

      2. expm1-define 41.9%

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

      3. log1p-define 48.1%

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

      4. unpow2 48.1%

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

      5. expm1-log1p-u 48.5%

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

      6. clear-num 48.5%

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

      7. un-div-inv 48.5%

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

   6. Applied egg-rr 48.5%

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

   7. Taylor expanded in t around 0 88.4%

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

      1. associate-/l* 89.3%

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

      2. *-commutative 89.3%

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

      3. associate-*r/ 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*l* 89.9%

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

      6. *-commutative 89.9%

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

   9. Simplified 89.9%

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

       1. associate-*l/ 88.5%

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

       2. pow-prod-down 88.4%

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

   11. Applied egg-rr 88.4%

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

   if 2.0000000000000001e300 < (*.f64 l l)

   1. Initial program 40.2%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in k around inf 56.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{\sin k\_m}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{{\left(k\_m \cdot \sin k\_m\right)}^{2}}{\cos k\_m \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 1.9e-14)
    (/ 2.0 (pow (* k_m (* (/ (sin k_m) l) (sqrt (/ t_m (cos k_m))))) 2.0))
    (if (<= k_m 4.1e+148)
      (/ 2.0 (* t_m (/ (pow (* k_m (sin k_m)) 2.0) (* (cos k_m) (pow l 2.0)))))
      (/
       2.0
       (*
        (pow (* (/ k_m t_m) (/ (pow t_m 1.5) l)) 2.0)
        (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = 2.0 / pow((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 4.1e+148) {
		tmp = 2.0 / (t_m * (pow((k_m * sin(k_m)), 2.0) / (cos(k_m) * pow(l, 2.0))));
	} else {
		tmp = 2.0 / (pow(((k_m / t_m) * (pow(t_m, 1.5) / l)), 2.0) * (sin(k_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-14) then
        tmp = 2.0d0 / ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else if (k_m <= 4.1d+148) then
        tmp = 2.0d0 / (t_m * (((k_m * sin(k_m)) ** 2.0d0) / (cos(k_m) * (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((k_m / t_m) * ((t_m ** 1.5d0) / l)) ** 2.0d0) * (sin(k_m) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-14) {
		tmp = 2.0 / Math.pow((k_m * ((Math.sin(k_m) / l) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 4.1e+148) {
		tmp = 2.0 / (t_m * (Math.pow((k_m * Math.sin(k_m)), 2.0) / (Math.cos(k_m) * Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / (Math.pow(((k_m / t_m) * (Math.pow(t_m, 1.5) / l)), 2.0) * (Math.sin(k_m) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.9e-14:
		tmp = 2.0 / math.pow((k_m * ((math.sin(k_m) / l) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	elif k_m <= 4.1e+148:
		tmp = 2.0 / (t_m * (math.pow((k_m * math.sin(k_m)), 2.0) / (math.cos(k_m) * math.pow(l, 2.0))))
	else:
		tmp = 2.0 / (math.pow(((k_m / t_m) * (math.pow(t_m, 1.5) / l)), 2.0) * (math.sin(k_m) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-14)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(sin(k_m) / l) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	elseif (k_m <= 4.1e+148)
		tmp = Float64(2.0 / Float64(t_m * Float64((Float64(k_m * sin(k_m)) ^ 2.0) / Float64(cos(k_m) * (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(k_m / t_m) * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(sin(k_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.9e-14)
		tmp = 2.0 / ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	elseif (k_m <= 4.1e+148)
		tmp = 2.0 / (t_m * (((k_m * sin(k_m)) ^ 2.0) / (cos(k_m) * (l ^ 2.0))));
	else
		tmp = 2.0 / ((((k_m / t_m) * ((t_m ^ 1.5) / l)) ^ 2.0) * (sin(k_m) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-14], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.1e+148], N[(2.0 / N[(t$95$m * N[(N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9000000000000001e-14

   1. Initial program 41.4%

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

      1. add-sqr-sqrt 20.7%

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

      2. pow2 20.7%

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

   4. Applied egg-rr 25.5%

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

   5. Taylor expanded in k around inf 49.5%

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

      1. associate-/l* 50.0%

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

      2. associate-*l* 49.2%

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

   7. Simplified 49.2%

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

   if 1.9000000000000001e-14 < k < 4.0999999999999998e148

   1. Initial program 16.2%

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

      1. associate-/r* 22.2%

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

      2. div-inv 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. add-exp-log 21.6%

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

      2. expm1-define 21.6%

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

      3. log1p-define 31.8%

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

      4. unpow2 31.8%

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

      5. expm1-log1p-u 32.5%

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

      6. clear-num 32.5%

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

      7. un-div-inv 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in t around 0 78.3%

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

      1. associate-/l* 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-*l* 81.9%

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

      6. *-commutative 81.9%

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

   9. Simplified 81.9%

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

       1. associate-*l/ 81.9%

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

       2. pow-prod-down 81.8%

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

   11. Applied egg-rr 81.8%

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

   if 4.0999999999999998e148 < k

   1. Initial program 40.5%

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

      1. add-sqr-sqrt 17.3%

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

      2. pow2 17.3%

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

   4. Applied egg-rr 14.2%

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

      1. associate-*r* 14.2%

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

      2. unpow-prod-down 14.2%

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

      3. pow2 14.2%

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

      4. add-sqr-sqrt 34.3%

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

   6. Applied egg-rr 34.3%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+211)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (/ 2.0 (pow (* (sqrt (/ t_m (cos k_m))) (/ (* k_m (sin k_m)) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+211) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+211) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+211) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * ((k_m * Math.sin(k_m)) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+211:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k_m))) * ((k_m * math.sin(k_m)) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+211)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(Float64(k_m * sin(k_m)) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+211)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+211], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999995e211

   1. Initial program 36.7%

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

      1. add-sqr-sqrt 17.8%

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

      2. pow2 17.8%

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

   4. Applied egg-rr 20.2%

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

   5. Taylor expanded in k around 0 37.4%

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

   if 4.9999999999999995e211 < (*.f64 l l)

   1. Initial program 40.5%

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

      1. add-sqr-sqrt 20.8%

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

      2. pow2 20.8%

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

   4. Applied egg-rr 28.2%

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

   5. Taylor expanded in k around inf 56.2%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot {\left(k\_m \cdot \frac{\frac{\sin k\_m}{\ell}}{\sqrt{\cos k\_m}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+211)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (/ 2.0 (* t_m (pow (* k_m (/ (/ (sin k_m) l) (sqrt (cos k_m)))) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+211) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / (t_m * pow((k_m * ((sin(k_m) / l) / sqrt(cos(k_m)))), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+211) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_m * ((k_m * ((sin(k_m) / l) / sqrt(cos(k_m)))) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+211) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / (t_m * Math.pow((k_m * ((Math.sin(k_m) / l) / Math.sqrt(Math.cos(k_m)))), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+211:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / (t_m * math.pow((k_m * ((math.sin(k_m) / l) / math.sqrt(math.cos(k_m)))), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+211)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_m * (Float64(k_m * Float64(Float64(sin(k_m) / l) / sqrt(cos(k_m)))) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+211)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / (t_m * ((k_m * ((sin(k_m) / l) / sqrt(cos(k_m)))) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+211], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] / N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999995e211

   1. Initial program 36.7%

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

      1. add-sqr-sqrt 17.8%

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

      2. pow2 17.8%

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

   4. Applied egg-rr 20.2%

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

   5. Taylor expanded in k around 0 37.4%

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

   if 4.9999999999999995e211 < (*.f64 l l)

   1. Initial program 40.5%

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

      1. associate-/r* 42.8%

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

      2. div-inv 42.9%

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

   4. Applied egg-rr 42.9%

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

      1. add-exp-log 42.6%

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

      2. expm1-define 42.6%

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

      3. log1p-define 42.8%

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

      4. unpow2 42.8%

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

      5. expm1-log1p-u 43.1%

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

      6. clear-num 43.1%

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

      7. un-div-inv 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in t around 0 64.6%

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

      1. associate-/l* 64.1%

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

      2. *-commutative 64.1%

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

      3. associate-*r/ 64.1%

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

      4. *-commutative 64.1%

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

      5. associate-*l* 65.8%

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

      6. *-commutative 65.8%

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

   9. Simplified 65.8%

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

       1. pow1 65.8%

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

   11. Applied egg-rr 71.4%

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

       1. unpow1 71.4%

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

       2. associate-/r* 71.5%

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

   13. Simplified 71.5%

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

Alternative 10: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k\_m}}{k\_m \cdot \frac{\sqrt{t\_m}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{t\_m} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 1.85e-118)
    (pow (/ (/ (sqrt 2.0) k_m) (* k_m (/ (sqrt t_m) l))) 2.0)
    (if (<= t_m 7e+76)
      (/
       2.0
       (/
        (* (/ k_m t_m) (* (* (sin k_m) (tan k_m)) (/ (/ (pow t_m 3.0) l) l)))
        (/ t_m k_m)))
      (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.85e-118) {
		tmp = pow(((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 7e+76) {
		tmp = 2.0 / (((k_m / t_m) * ((sin(k_m) * tan(k_m)) * ((pow(t_m, 3.0) / l) / l))) / (t_m / k_m));
	} else {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.85d-118) then
        tmp = ((sqrt(2.0d0) / k_m) / (k_m * (sqrt(t_m) / l))) ** 2.0d0
    else if (t_m <= 7d+76) then
        tmp = 2.0d0 / (((k_m / t_m) * ((sin(k_m) * tan(k_m)) * (((t_m ** 3.0d0) / l) / l))) / (t_m / k_m))
    else
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.85e-118) {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 7e+76) {
		tmp = 2.0 / (((k_m / t_m) * ((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 3.0) / l) / l))) / (t_m / k_m));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.85e-118:
		tmp = math.pow(((math.sqrt(2.0) / k_m) / (k_m * (math.sqrt(t_m) / l))), 2.0)
	elif t_m <= 7e+76:
		tmp = 2.0 / (((k_m / t_m) * ((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 3.0) / l) / l))) / (t_m / k_m))
	else:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-118)
		tmp = Float64(Float64(sqrt(2.0) / k_m) / Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0;
	elseif (t_m <= 7e+76)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t_m) * Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 3.0) / l) / l))) / Float64(t_m / k_m)));
	else
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.85e-118)
		tmp = ((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))) ^ 2.0;
	elseif (t_m <= 7e+76)
		tmp = 2.0 / (((k_m / t_m) * ((sin(k_m) * tan(k_m)) * (((t_m ^ 3.0) / l) / l))) / (t_m / k_m));
	else
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-118], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 7e+76], N[(2.0 / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.85000000000000007e-118

   1. Initial program 34.0%

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

   3. Taylor expanded in t around 0 72.5%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

   5. Simplified 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. associate-/r* 73.0%

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

      3. associate-/l* 73.1%

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

   7. Applied egg-rr 73.1%

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

   8. Taylor expanded in k around 0 62.9%

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

      1. associate-/l* 61.9%

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

   10. Simplified 61.9%

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

       1. add-sqr-sqrt 34.3%

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

       2. pow2 34.3%

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

       3. sqrt-div 13.5%

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

       4. sqrt-div 13.5%

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

       5. sqrt-pow1 13.5%

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

       6. metadata-eval 13.5%

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

       7. pow1 13.5%

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

       8. sqrt-prod 13.5%

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

       9. sqrt-pow1 14.9%

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

       10. metadata-eval 14.9%

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

       11. pow1 14.9%

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

       12. sqrt-div 13.0%

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

       13. sqrt-pow1 15.8%

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

       14. metadata-eval 15.8%

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

       15. pow1 15.8%

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

   12. Applied egg-rr 15.8%

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

   if 1.85000000000000007e-118 < t < 7.00000000000000001e76

   1. Initial program 74.4%

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

      1. associate-/r* 74.4%

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

      2. div-inv 74.4%

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

   4. Applied egg-rr 74.4%

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

      1. add-exp-log 73.8%

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

      2. expm1-define 73.8%

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

      3. log1p-define 76.1%

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

      4. unpow2 76.1%

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

      5. expm1-log1p-u 76.7%

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

      6. clear-num 76.7%

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

      7. un-div-inv 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*r/ 77.1%

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

      2. associate-*l* 77.2%

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

      3. un-div-inv 77.2%

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

   8. Applied egg-rr 77.2%

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

   if 7.00000000000000001e76 < t

   1. Initial program 11.1%

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

      1. add-sqr-sqrt 5.4%

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

      2. pow2 5.4%

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

   4. Applied egg-rr 40.5%

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

   5. Taylor expanded in k around 0 85.6%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k}}{k \cdot \frac{\sqrt{t}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{t} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k\_m}}{k\_m \cdot \frac{\sqrt{t\_m}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)\right) \cdot \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 1.85e-118)
    (pow (/ (/ (sqrt 2.0) k_m) (* k_m (/ (sqrt t_m) l))) 2.0)
    (if (<= t_m 1.2e+65)
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (/ (pow t_m 3.0) l) l)))
        (/ (/ k_m t_m) (/ t_m k_m))))
      (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.85e-118) {
		tmp = pow(((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 1.2e+65) {
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * ((pow(t_m, 3.0) / l) / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.85d-118) then
        tmp = ((sqrt(2.0d0) / k_m) / (k_m * (sqrt(t_m) / l))) ** 2.0d0
    else if (t_m <= 1.2d+65) then
        tmp = 2.0d0 / ((tan(k_m) * (sin(k_m) * (((t_m ** 3.0d0) / l) / l))) * ((k_m / t_m) / (t_m / k_m)))
    else
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.85e-118) {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 1.2e+65) {
		tmp = 2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * ((Math.pow(t_m, 3.0) / l) / l))) * ((k_m / t_m) / (t_m / k_m)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.85e-118:
		tmp = math.pow(((math.sqrt(2.0) / k_m) / (k_m * (math.sqrt(t_m) / l))), 2.0)
	elif t_m <= 1.2e+65:
		tmp = 2.0 / ((math.tan(k_m) * (math.sin(k_m) * ((math.pow(t_m, 3.0) / l) / l))) * ((k_m / t_m) / (t_m / k_m)))
	else:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-118)
		tmp = Float64(Float64(sqrt(2.0) / k_m) / Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0;
	elseif (t_m <= 1.2e+65)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64(Float64((t_m ^ 3.0) / l) / l))) * Float64(Float64(k_m / t_m) / Float64(t_m / k_m))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.85e-118)
		tmp = ((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))) ^ 2.0;
	elseif (t_m <= 1.2e+65)
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * (((t_m ^ 3.0) / l) / l))) * ((k_m / t_m) / (t_m / k_m)));
	else
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-118], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 1.2e+65], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.85000000000000007e-118

   1. Initial program 34.0%

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

   3. Taylor expanded in t around 0 72.5%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

   5. Simplified 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. associate-/r* 73.0%

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

      3. associate-/l* 73.1%

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

   7. Applied egg-rr 73.1%

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

   8. Taylor expanded in k around 0 62.9%

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

      1. associate-/l* 61.9%

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

   10. Simplified 61.9%

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

       1. add-sqr-sqrt 34.3%

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

       2. pow2 34.3%

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

       3. sqrt-div 13.5%

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

       4. sqrt-div 13.5%

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

       5. sqrt-pow1 13.5%

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

       6. metadata-eval 13.5%

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

       7. pow1 13.5%

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

       8. sqrt-prod 13.5%

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

       9. sqrt-pow1 14.9%

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

       10. metadata-eval 14.9%

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

       11. pow1 14.9%

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

       12. sqrt-div 13.0%

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

       13. sqrt-pow1 15.8%

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

       14. metadata-eval 15.8%

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

       15. pow1 15.8%

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

   12. Applied egg-rr 15.8%

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

   if 1.85000000000000007e-118 < t < 1.2000000000000001e65

   1. Initial program 76.1%

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

      1. associate-/r* 76.1%

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

      2. div-inv 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. add-exp-log 75.5%

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

      2. expm1-define 75.5%

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

      3. log1p-define 77.8%

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

      4. unpow2 77.8%

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

      5. expm1-log1p-u 78.4%

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

      6. clear-num 78.4%

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

      7. un-div-inv 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. un-div-inv 78.3%

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

   8. Applied egg-rr 78.3%

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

   if 1.2000000000000001e65 < t

   1. Initial program 16.9%

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

      1. add-sqr-sqrt 9.5%

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

      2. pow2 9.5%

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

   4. Applied egg-rr 40.6%

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

   5. Taylor expanded in k around 0 85.1%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k}}{k \cdot \frac{\sqrt{t}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.45 \cdot 10^{+71}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k\_m}^{2}}{t\_m} + 2 \cdot \frac{1}{t\_m}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k\_m}}{k\_m \cdot \frac{\sqrt{t\_m}}{\ell}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.2e+14)
    (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))
    (if (<= k_m 1.45e+71)
      (*
       (* l l)
       (/
        (+ (* -0.3333333333333333 (/ (pow k_m 2.0) t_m)) (* 2.0 (/ 1.0 t_m)))
        (pow k_m 4.0)))
      (pow (/ (/ (sqrt 2.0) k_m) (* k_m (/ (sqrt t_m) l))) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e+14) {
		tmp = 2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if (k_m <= 1.45e+71) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / pow(k_m, 4.0));
	} else {
		tmp = pow(((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.2d+14) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if (k_m <= 1.45d+71) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k_m ** 2.0d0) / t_m)) + (2.0d0 * (1.0d0 / t_m))) / (k_m ** 4.0d0))
    else
        tmp = ((sqrt(2.0d0) / k_m) / (k_m * (sqrt(t_m) / l))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e+14) {
		tmp = 2.0 / Math.pow(((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if (k_m <= 1.45e+71) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / Math.pow(k_m, 4.0));
	} else {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / (k_m * (Math.sqrt(t_m) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 6.2e+14:
		tmp = 2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif k_m <= 1.45e+71:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / math.pow(k_m, 4.0))
	else:
		tmp = math.pow(((math.sqrt(2.0) / k_m) / (k_m * (math.sqrt(t_m) / l))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 6.2e+14)
		tmp = Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (k_m <= 1.45e+71)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k_m ^ 2.0) / t_m)) + Float64(2.0 * Float64(1.0 / t_m))) / (k_m ^ 4.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / k_m) / Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.2e+14)
		tmp = 2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif (k_m <= 1.45e+71)
		tmp = (l * l) * (((-0.3333333333333333 * ((k_m ^ 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / (k_m ^ 4.0));
	else
		tmp = ((sqrt(2.0) / k_m) / (k_m * (sqrt(t_m) / l))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.2e+14], N[(2.0 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.45e+71], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.2e14

   1. Initial program 40.3%

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

      1. add-sqr-sqrt 20.2%

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

      2. pow2 20.2%

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

   4. Applied egg-rr 26.4%

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

   5. Taylor expanded in k around 0 39.4%

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

   if 6.2e14 < k < 1.45000000000000004e71

   1. Initial program 23.3%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in k around 0 53.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 1.45000000000000004e71 < k

   1. Initial program 33.3%

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

   3. Taylor expanded in t around 0 56.2%

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

      1. associate-/l* 54.4%

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

      2. *-commutative 54.4%

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

   5. Simplified 54.4%

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

      1. *-un-lft-identity 54.4%

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

      2. associate-/r* 54.4%

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

      3. associate-/l* 54.4%

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

   7. Applied egg-rr 54.4%

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

   8. Taylor expanded in k around 0 50.3%

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

      1. associate-/l* 50.4%

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

   10. Simplified 50.4%

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

       1. add-sqr-sqrt 50.2%

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

       2. pow2 50.2%

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

       3. sqrt-div 25.2%

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

       4. sqrt-div 25.2%

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

       5. sqrt-pow1 25.2%

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

       6. metadata-eval 25.2%

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

       7. pow1 25.2%

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

       8. sqrt-prod 25.2%

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

       9. sqrt-pow1 25.7%

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

       10. metadata-eval 25.7%

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

       11. pow1 25.7%

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

       12. sqrt-div 25.3%

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

       13. sqrt-pow1 23.6%

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

       14. metadata-eval 23.6%

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

       15. pow1 23.6%

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

   12. Applied egg-rr 23.6%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+71}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k}}{k \cdot \frac{\sqrt{t}}{\ell}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.1% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* (/ (pow k_m 2.0) l) (sqrt t_m)) 2.0))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / pow(((pow(k_m, 2.0) / l) * sqrt(t_m)), 2.0));
}

Fortran

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

Java

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

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / math.pow(((math.pow(k_m, 2.0) / l) * math.sqrt(t_m)), 2.0))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / (Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((((k_m ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. add-sqr-sqrt 18.9%

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

   2. pow2 18.9%

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

4. Applied egg-rr 23.1%

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

5. Taylor expanded in k around 0 34.5%

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

Alternative 14: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k\_m}^{2}}}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* (/ k_m t_m) (* k_m (/ (pow t_m 1.5) l))) 2.0))
    (/ (/ 2.0 (pow k_m 2.0)) (* (* k_m k_m) (/ t_m (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow(((k_m / t_m) * (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (2.0 / pow(k_m, 2.0)) / ((k_m * k_m) * (t_m / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / (((k_m / t_m) * (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = (2.0d0 / (k_m ** 2.0d0)) / ((k_m * k_m) * (t_m / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow(((k_m / t_m) * (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (2.0 / Math.pow(k_m, 2.0)) / ((k_m * k_m) * (t_m / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow(((k_m / t_m) * (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = (2.0 / math.pow(k_m, 2.0)) / ((k_m * k_m) * (t_m / math.pow(l, 2.0)))
	return t_s * tmp

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 25.9%

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

      1. add-sqr-sqrt 9.3%

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

      2. pow2 9.3%

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

   4. Applied egg-rr 13.2%

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

   5. Taylor expanded in k around 0 20.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.3%

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

   3. Taylor expanded in t around 0 78.8%

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

      1. associate-/l* 79.4%

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

      2. *-commutative 79.4%

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

   5. Simplified 79.4%

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

      1. *-un-lft-identity 79.4%

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

      2. associate-/r* 79.6%

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

      3. associate-/l* 79.7%

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

   7. Applied egg-rr 79.7%

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

   8. Taylor expanded in k around 0 67.3%

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

      1. associate-/l* 65.8%

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

   10. Simplified 65.8%

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

       1. unpow2 79.4%

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

   12. Applied egg-rr 65.8%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2}}}{\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* (/ k_m t_m) (* k_m (/ (pow t_m 1.5) l))) 2.0))
    (* (* l l) (* (/ 2.0 t_m) (pow k_m -4.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow(((k_m / t_m) * (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (l * 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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / (((k_m / t_m) * (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = (l * 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 ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow(((k_m / t_m) * (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = (l * 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 (l * l) <= 0.0:
		tmp = 2.0 / math.pow(((k_m / t_m) * (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = (l * 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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(Float64(k_m / t_m) * Float64(k_m * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / (((k_m / t_m) * (k_m * ((t_m ^ 1.5) / l))) ^ 2.0);
	else
		tmp = (l * 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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 25.9%

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

      1. add-sqr-sqrt 9.3%

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

      2. pow2 9.3%

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

   4. Applied egg-rr 13.2%

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

   5. Taylor expanded in k around 0 20.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.3%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in k around 0 64.5%

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

      1. *-un-lft-identity 64.5%

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

      2. associate-/r* 65.0%

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

   7. Applied egg-rr 65.0%

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

   8. Taylor expanded in k around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/r* 65.0%

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

   10. Simplified 65.0%

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

       1. associate-/l/ 64.5%

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

       2. *-un-lft-identity 64.5%

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

       3. associate-/l/ 65.0%

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

       4. div-inv 65.0%

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

       5. pow-flip 65.0%

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

       6. metadata-eval 65.0%

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

   12. Applied egg-rr 65.0%

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

       1. *-lft-identity 65.0%

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

   14. Simplified 65.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.6% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* t_m (/ (pow k_m 4.0) (pow l 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. associate-/r* 42.9%

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

   2. div-inv 42.9%

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

4. Applied egg-rr 42.9%

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

   1. add-exp-log 42.6%

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

   2. expm1-define 42.6%

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

   3. log1p-define 47.9%

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

   4. unpow2 47.9%

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

   5. expm1-log1p-u 48.1%

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

   6. clear-num 48.1%

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

   7. un-div-inv 48.1%

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

6. Applied egg-rr 48.1%

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

7. Taylor expanded in t around 0 74.0%

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

   1. associate-/l* 74.5%

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

   2. *-commutative 74.5%

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

   3. associate-*r/ 74.6%

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

   4. *-commutative 74.6%

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

   5. associate-*l* 74.9%

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

   6. *-commutative 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in k around 0 63.1%

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

Alternative 17: 63.3% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (* (/ 2.0 t_m) (pow k_m -4.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Simplified 42.9%

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

5. Taylor expanded in k around 0 62.7%

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

   1. *-un-lft-identity 62.7%

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

   2. associate-/r* 63.1%

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

7. Applied egg-rr 63.1%

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

8. Taylor expanded in k around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-/r* 63.0%

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

10. Simplified 63.0%

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

    1. associate-/l/ 62.7%

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

    2. *-un-lft-identity 62.7%

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

    3. associate-/l/ 63.0%

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

    4. div-inv 63.1%

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

    5. pow-flip 63.1%

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

    6. metadata-eval 63.1%

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

12. Applied egg-rr 63.1%

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

    1. *-lft-identity 63.1%

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

14. Simplified 63.1%

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

15. Final simplification 63.1%

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

Alternative 18: 63.3% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ (/ 2.0 (pow k_m 4.0)) t_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Simplified 42.9%

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

5. Taylor expanded in k around 0 62.7%

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

   1. *-un-lft-identity 62.7%

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

   2. associate-/r* 63.1%

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

7. Applied egg-rr 63.1%

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

8. Taylor expanded in k around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-/r* 63.0%

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

10. Simplified 63.0%

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

    1. *-un-lft-identity 63.0%

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

    2. associate-/l/ 62.7%

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

    3. associate-/r* 63.1%

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

12. Applied egg-rr 63.1%

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

13. Final simplification 63.1%

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

Alternative 19: 63.4% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Simplified 42.9%

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

5. Taylor expanded in k around 0 62.7%

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

6. Final simplification 62.7%

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

Reproduce

herbie shell --seed 2024112 
(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))))