Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 78.6%

Time: 31.9s

Alternatives: 22

Speedup: 2.0×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {\sin k}^{2}\\ t_3 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{t\_3}{{\left(k \cdot \sqrt{t\_2}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t\_m}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_3}{t\_2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (pow (sin k) 2.0))) (t_3 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 1.02e-60)
      (* 2.0 (/ t_3 (pow (* k (sqrt t_2)) 2.0)))
      (if (<= t_m 2.85e+203)
        (/
         (/
          2.0
          (*
           (* (pow t_m 1.5) (* (sin k) (/ 1.0 l)))
           (/ (pow t_m 1.5) (/ l (tan k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_3 t_2))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * pow(sin(k), 2.0);
	double t_3 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 1.02e-60) {
		tmp = 2.0 * (t_3 / pow((k * sqrt(t_2)), 2.0));
	} else if (t_m <= 2.85e+203) {
		tmp = (2.0 / ((pow(t_m, 1.5) * (sin(k) * (1.0 / l))) * (pow(t_m, 1.5) / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_3 / t_2));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * (sin(k) ** 2.0d0)
    t_3 = (l ** 2.0d0) * cos(k)
    if (t_m <= 1.02d-60) then
        tmp = 2.0d0 * (t_3 / ((k * sqrt(t_2)) ** 2.0d0))
    else if (t_m <= 2.85d+203) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) * (sin(k) * (1.0d0 / l))) * ((t_m ** 1.5d0) / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_3 / t_2))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = t_m * Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 1.02e-60) {
		tmp = 2.0 * (t_3 / Math.pow((k * Math.sqrt(t_2)), 2.0));
	} else if (t_m <= 2.85e+203) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) * (Math.sin(k) * (1.0 / l))) * (Math.pow(t_m, 1.5) / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_3 / t_2));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.pow(math.sin(k), 2.0)
	t_3 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 1.02e-60:
		tmp = 2.0 * (t_3 / math.pow((k * math.sqrt(t_2)), 2.0))
	elif t_m <= 2.85e+203:
		tmp = (2.0 / ((math.pow(t_m, 1.5) * (math.sin(k) * (1.0 / l))) * (math.pow(t_m, 1.5) / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_3 / t_2))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * (sin(k) ^ 2.0))
	t_3 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 1.02e-60)
		tmp = Float64(2.0 * Float64(t_3 / (Float64(k * sqrt(t_2)) ^ 2.0)));
	elseif (t_m <= 2.85e+203)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) * Float64(sin(k) * Float64(1.0 / l))) * Float64((t_m ^ 1.5) / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_3 / t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (sin(k) ^ 2.0);
	t_3 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 1.02e-60)
		tmp = 2.0 * (t_3 / ((k * sqrt(t_2)) ^ 2.0));
	elseif (t_m <= 2.85e+203)
		tmp = (2.0 / (((t_m ^ 1.5) * (sin(k) * (1.0 / l))) * ((t_m ^ 1.5) / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_3 / t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.02e-60], N[(2.0 * N[(t$95$3 / N[Power[N[(k * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+203], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.01999999999999994e-60

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-/r* 35.7%

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

      3. sub-neg 35.7%

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

      4. distribute-rgt-in 30.9%

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

      5. unpow2 30.9%

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

      6. times-frac 21.2%

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

      7. sqr-neg 21.2%

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

      8. times-frac 30.9%

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

      9. unpow2 30.9%

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

      10. distribute-rgt-in 35.7%

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

      11. +-commutative 35.7%

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

      12. associate-+l+ 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. add-sqr-sqrt 10.8%

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

      3. unpow2 10.8%

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

      4. pow-prod-down 12.9%

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

   7. Applied egg-rr 12.9%

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

   if 1.01999999999999994e-60 < t < 2.85e203

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 46.6%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.7%

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

      3. times-frac 64.9%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 85.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. div-inv 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.6%

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

   10. Simplified 85.6%

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

   if 2.85e203 < t

   1. Initial program 3.7%

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

   3. Taylor expanded in t around 0 70.8%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \sqrt{t \cdot {\sin k}^{2}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-61}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\left(k \cdot \sqrt{t\_2}\right)}^{-2}\right)\right)\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t\_m}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t\_2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (pow (sin k) 2.0))))
   (*
    t_s
    (if (<= t_m 3.3e-61)
      (* 2.0 (* (pow l 2.0) (* (cos k) (pow (* k (sqrt t_2)) -2.0))))
      (if (<= t_m 3.3e+203)
        (/
         (/
          2.0
          (*
           (* (pow t_m 1.5) (* (sin k) (/ 1.0 l)))
           (/ (pow t_m 1.5) (/ l (tan k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ (* (pow l 2.0) (cos k)) t_2))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 3.3e-61) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) * pow((k * sqrt(t_2)), -2.0)));
	} else if (t_m <= 3.3e+203) {
		tmp = (2.0 / ((pow(t_m, 1.5) * (sin(k) * (1.0 / l))) * (pow(t_m, 1.5) / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / ((pow(l, 2.0) * cos(k)) / t_2));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sin(k) ** 2.0d0)
    if (t_m <= 3.3d-61) then
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k) * ((k * sqrt(t_2)) ** (-2.0d0))))
    else if (t_m <= 3.3d+203) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) * (sin(k) * (1.0d0 / l))) * ((t_m ** 1.5d0) / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (((l ** 2.0d0) * cos(k)) / t_2))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = t_m * Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 3.3e-61) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * Math.pow((k * Math.sqrt(t_2)), -2.0)));
	} else if (t_m <= 3.3e+203) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) * (Math.sin(k) * (1.0 / l))) * (Math.pow(t_m, 1.5) / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k)) / t_2));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 3.3e-61:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k) * math.pow((k * math.sqrt(t_2)), -2.0)))
	elif t_m <= 3.3e+203:
		tmp = (2.0 / ((math.pow(t_m, 1.5) * (math.sin(k) * (1.0 / l))) * (math.pow(t_m, 1.5) / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / ((math.pow(l, 2.0) * math.cos(k)) / t_2))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * (sin(k) ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.3e-61)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * (Float64(k * sqrt(t_2)) ^ -2.0))));
	elseif (t_m <= 3.3e+203)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) * Float64(sin(k) * Float64(1.0 / l))) * Float64((t_m ^ 1.5) / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k)) / t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (sin(k) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.3e-61)
		tmp = 2.0 * ((l ^ 2.0) * (cos(k) * ((k * sqrt(t_2)) ^ -2.0)));
	elseif (t_m <= 3.3e+203)
		tmp = (2.0 / (((t_m ^ 1.5) * (sin(k) * (1.0 / l))) * ((t_m ^ 1.5) / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (((l ^ 2.0) * cos(k)) / t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.3e-61], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[(k * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+203], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999996e-61

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-/r* 35.7%

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

      3. sub-neg 35.7%

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

      4. distribute-rgt-in 30.9%

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

      5. unpow2 30.9%

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

      6. times-frac 21.2%

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

      7. sqr-neg 21.2%

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

      8. times-frac 30.9%

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

      9. unpow2 30.9%

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

      10. distribute-rgt-in 35.7%

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

      11. +-commutative 35.7%

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

      12. associate-+l+ 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. add-sqr-sqrt 10.8%

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

      3. unpow2 10.8%

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

      4. pow-prod-down 12.9%

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

   7. Applied egg-rr 12.9%

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

      1. div-inv 12.9%

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

      2. pow-flip 12.9%

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

      3. *-commutative 12.9%

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

      4. metadata-eval 12.9%

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

   9. Applied egg-rr 12.9%

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

       1. associate-*l* 12.9%

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

   11. Simplified 12.9%

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

   if 3.29999999999999996e-61 < t < 3.29999999999999989e203

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 46.6%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.7%

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

      3. times-frac 64.9%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 85.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. div-inv 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.6%

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

   10. Simplified 85.6%

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

   if 3.29999999999999989e203 < t

   1. Initial program 3.7%

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

   3. Taylor expanded in t around 0 70.8%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-61}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\left(k \cdot \sqrt{t \cdot {\sin k}^{2}}\right)}^{-2}\right)\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t\_m}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {\sin k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 2.05e-60)
      (* 2.0 (/ t_2 (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
      (if (<= t_m 2.95e+203)
        (/
         (/
          2.0
          (*
           (* (pow t_m 1.5) (* (sin k) (/ 1.0 l)))
           (/ (pow t_m 1.5) (/ l (tan k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow (sin k) 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 2.05e-60) {
		tmp = 2.0 * (t_2 / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else if (t_m <= 2.95e+203) {
		tmp = (2.0 / ((pow(t_m, 1.5) * (sin(k) * (1.0 / l))) * (pow(t_m, 1.5) / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 2.05d-60) then
        tmp = 2.0d0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 2.95d+203) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) * (sin(k) * (1.0d0 / l))) * ((t_m ** 1.5d0) / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 2.05e-60) {
		tmp = 2.0 * (t_2 / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 2.95e+203) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) * (Math.sin(k) * (1.0 / l))) * (Math.pow(t_m, 1.5) / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 2.05e-60:
		tmp = 2.0 * (t_2 / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	elif t_m <= 2.95e+203:
		tmp = (2.0 / ((math.pow(t_m, 1.5) * (math.sin(k) * (1.0 / l))) * (math.pow(t_m, 1.5) / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 2.05e-60)
		tmp = Float64(2.0 * Float64(t_2 / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 2.95e+203)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) * Float64(sin(k) * Float64(1.0 / l))) * Float64((t_m ^ 1.5) / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 2.05e-60)
		tmp = 2.0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 2.95e+203)
		tmp = (2.0 / (((t_m ^ 1.5) * (sin(k) * (1.0 / l))) * ((t_m ^ 1.5) / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-60], N[(2.0 * N[(t$95$2 / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.95e+203], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.05000000000000006e-60

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-/r* 35.7%

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

      3. sub-neg 35.7%

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

      4. distribute-rgt-in 30.9%

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

      5. unpow2 30.9%

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

      6. times-frac 21.2%

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

      7. sqr-neg 21.2%

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

      8. times-frac 30.9%

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

      9. unpow2 30.9%

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

      10. distribute-rgt-in 35.7%

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

      11. +-commutative 35.7%

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

      12. associate-+l+ 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. add-sqr-sqrt 10.8%

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

      3. unpow2 10.8%

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

      4. pow-prod-down 12.9%

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

   7. Applied egg-rr 12.9%

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

   8. Taylor expanded in k around inf 12.9%

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

   if 2.05000000000000006e-60 < t < 2.94999999999999986e203

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 46.6%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.7%

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

      3. times-frac 64.9%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 85.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. div-inv 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.6%

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

   10. Simplified 85.6%

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

   if 2.94999999999999986e203 < t

   1. Initial program 3.7%

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

   3. Taylor expanded in t around 0 70.8%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\left({t}^{1.5} \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right) \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.6 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.65 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}} \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {\sin k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 8.6e-59)
      (* 2.0 (/ t_2 (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
      (if (<= t_m 2.65e+203)
        (/
         (/
          2.0
          (* (/ (pow t_m 1.5) (/ l (tan k))) (/ (pow t_m 1.5) (/ l (sin k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow (sin k) 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 8.6e-59) {
		tmp = 2.0 * (t_2 / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else if (t_m <= 2.65e+203) {
		tmp = (2.0 / ((pow(t_m, 1.5) / (l / tan(k))) * (pow(t_m, 1.5) / (l / sin(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 8.6d-59) then
        tmp = 2.0d0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 2.65d+203) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) / (l / tan(k))) * ((t_m ** 1.5d0) / (l / sin(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 8.6e-59) {
		tmp = 2.0 * (t_2 / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 2.65e+203) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) / (l / Math.tan(k))) * (Math.pow(t_m, 1.5) / (l / Math.sin(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 8.6e-59:
		tmp = 2.0 * (t_2 / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	elif t_m <= 2.65e+203:
		tmp = (2.0 / ((math.pow(t_m, 1.5) / (l / math.tan(k))) * (math.pow(t_m, 1.5) / (l / math.sin(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 8.6e-59)
		tmp = Float64(2.0 * Float64(t_2 / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 2.65e+203)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) / Float64(l / tan(k))) * Float64((t_m ^ 1.5) / Float64(l / sin(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 8.6e-59)
		tmp = 2.0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 2.65e+203)
		tmp = (2.0 / (((t_m ^ 1.5) / (l / tan(k))) * ((t_m ^ 1.5) / (l / sin(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.6e-59], N[(2.0 * N[(t$95$2 / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.65e+203], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.6000000000000006e-59

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-/r* 35.7%

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

      3. sub-neg 35.7%

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

      4. distribute-rgt-in 30.9%

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

      5. unpow2 30.9%

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

      6. times-frac 21.2%

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

      7. sqr-neg 21.2%

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

      8. times-frac 30.9%

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

      9. unpow2 30.9%

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

      10. distribute-rgt-in 35.7%

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

      11. +-commutative 35.7%

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

      12. associate-+l+ 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. add-sqr-sqrt 10.8%

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

      3. unpow2 10.8%

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

      4. pow-prod-down 12.9%

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

   7. Applied egg-rr 12.9%

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

   8. Taylor expanded in k around inf 12.9%

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

   if 8.6000000000000006e-59 < t < 2.64999999999999994e203

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 46.6%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.7%

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

      3. times-frac 64.9%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 85.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

   if 2.64999999999999994e203 < t

   1. Initial program 3.7%

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

   3. Taylor expanded in t around 0 70.8%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\tan k}} \cdot \frac{{t}^{1.5}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.45 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)}^{2}}{\cos k}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {\sin k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 4.2e-57)
      (* 2.0 (/ t_2 (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
      (if (<= t_m 2.45e+203)
        (/
         (/ 2.0 (/ (pow (/ (* (sin k) (pow t_m 1.5)) l) 2.0) (cos k)))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow (sin k) 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 4.2e-57) {
		tmp = 2.0 * (t_2 / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else if (t_m <= 2.45e+203) {
		tmp = (2.0 / (pow(((sin(k) * pow(t_m, 1.5)) / l), 2.0) / cos(k))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 4.2d-57) then
        tmp = 2.0d0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 2.45d+203) then
        tmp = (2.0d0 / ((((sin(k) * (t_m ** 1.5d0)) / l) ** 2.0d0) / cos(k))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 4.2e-57) {
		tmp = 2.0 * (t_2 / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 2.45e+203) {
		tmp = (2.0 / (Math.pow(((Math.sin(k) * Math.pow(t_m, 1.5)) / l), 2.0) / Math.cos(k))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 4.2e-57:
		tmp = 2.0 * (t_2 / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	elif t_m <= 2.45e+203:
		tmp = (2.0 / (math.pow(((math.sin(k) * math.pow(t_m, 1.5)) / l), 2.0) / math.cos(k))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 4.2e-57)
		tmp = Float64(2.0 * Float64(t_2 / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 2.45e+203)
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(sin(k) * (t_m ^ 1.5)) / l) ^ 2.0) / cos(k))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 4.2e-57)
		tmp = 2.0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 2.45e+203)
		tmp = (2.0 / ((((sin(k) * (t_m ^ 1.5)) / l) ^ 2.0) / cos(k))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-57], N[(2.0 * N[(t$95$2 / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.45e+203], N[(N[(2.0 / N[(N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.1999999999999999e-57

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-/r* 35.7%

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

      3. sub-neg 35.7%

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

      4. distribute-rgt-in 30.9%

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

      5. unpow2 30.9%

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

      6. times-frac 21.2%

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

      7. sqr-neg 21.2%

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

      8. times-frac 30.9%

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

      9. unpow2 30.9%

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

      10. distribute-rgt-in 35.7%

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

      11. +-commutative 35.7%

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

      12. associate-+l+ 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. add-sqr-sqrt 10.8%

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

      3. unpow2 10.8%

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

      4. pow-prod-down 12.9%

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

   7. Applied egg-rr 12.9%

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

   8. Taylor expanded in k around inf 12.9%

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

   if 4.1999999999999999e-57 < t < 2.4499999999999999e203

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 46.6%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. *-commutative 57.9%

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

      2. times-frac 64.7%

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

   6. Applied egg-rr 64.7%

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

   7. Taylor expanded in t around 0 57.8%

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

      1. times-frac 57.9%

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

      2. associate-*r/ 57.9%

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

      3. metadata-eval 57.9%

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

      4. pow-sqr 57.9%

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

      5. unpow2 57.9%

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

      6. times-frac 81.0%

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

      7. unpow2 81.0%

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

      8. swap-sqr 85.7%

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

      9. unpow1 85.7%

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

      10. pow-plus 85.7%

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

      11. *-commutative 85.7%

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

      12. associate-*r/ 85.6%

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

      13. metadata-eval 85.6%

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

   9. Simplified 85.6%

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

   if 2.4499999999999999e203 < t

   1. Initial program 3.7%

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

   3. Taylor expanded in t around 0 70.8%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{\sin k \cdot {t}^{1.5}}{\ell}\right)}^{2}}{\cos k}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-74)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
    (if (<= t_m 7.4e+58)
      (/
       (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ t_m (/ l (tan k)))))
       (pow (/ k t_m) 2.0))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (cos k) (* t_m (pow (sin k) 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-74) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else if (t_m <= 7.4e+58) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.4d-74) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 7.4d+58) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 3.4e-74) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 7.4e+58) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * (t_m / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.4e-74:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	elif t_m <= 7.4e+58:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * (t_m / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e-74)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 7.4e+58)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(t_m / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.4e-74)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 7.4e+58)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-74], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+58], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.4000000000000001e-74

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. add-sqr-sqrt 10.3%

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

      3. unpow2 10.3%

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

      4. pow-prod-down 12.4%

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

   7. Applied egg-rr 12.4%

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

   8. Taylor expanded in k around inf 12.4%

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

   if 3.4000000000000001e-74 < t < 7.4000000000000004e58

   1. Initial program 59.0%

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

      1. associate-/r* 59.0%

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

      2. associate-*l* 59.0%

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

      3. associate-*l/ 59.0%

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

      4. associate-/l* 59.0%

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

      5. +-commutative 59.0%

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

      6. unpow2 59.0%

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

      7. sqr-neg 59.0%

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

      8. distribute-frac-neg 59.0%

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

      9. distribute-frac-neg 59.0%

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

      10. unpow2 59.0%

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

      11. associate--l+ 59.4%

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

      12. metadata-eval 59.4%

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

      13. +-rgt-identity 59.4%

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

      14. unpow2 59.4%

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

      15. distribute-frac-neg 59.4%

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

      16. distribute-frac-neg 59.4%

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

      17. sqr-neg 59.4%

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

      18. unpow2 59.4%

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

   3. Simplified 59.4%

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

      1. unpow3 59.4%

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

      2. times-frac 76.4%

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

      3. times-frac 76.5%

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

      4. pow2 76.5%

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

   6. Applied egg-rr 76.5%

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

   if 7.4000000000000004e58 < t

   1. Initial program 22.2%

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

      1. associate-*l* 22.2%

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

      2. associate-/r* 22.2%

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

      3. sub-neg 22.2%

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

      4. distribute-rgt-in 22.2%

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

      5. unpow2 22.2%

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

      6. times-frac 14.8%

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

      7. sqr-neg 14.8%

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

      8. times-frac 22.2%

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

      9. unpow2 22.2%

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

      10. distribute-rgt-in 22.2%

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

      11. +-commutative 22.2%

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

      12. associate-+l+ 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. times-frac 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {\sin k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 2.9e-74)
      (* 2.0 (/ t_2 (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
      (if (<= t_m 1.2e+123)
        (/
         (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ t_m (/ l (tan k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow (sin k) 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 2.9e-74) {
		tmp = 2.0 * (t_2 / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else if (t_m <= 1.2e+123) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 2.9d-74) then
        tmp = 2.0d0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 1.2d+123) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 2.9e-74) {
		tmp = 2.0 * (t_2 / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 1.2e+123) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * (t_m / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 2.9e-74:
		tmp = 2.0 * (t_2 / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	elif t_m <= 1.2e+123:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * (t_m / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 2.9e-74)
		tmp = Float64(2.0 * Float64(t_2 / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 1.2e+123)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(t_m / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 2.9e-74)
		tmp = 2.0 * (t_2 / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 1.2e+123)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-74], N[(2.0 * N[(t$95$2 / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+123], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9e-74

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. add-sqr-sqrt 10.3%

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

      3. unpow2 10.3%

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

      4. pow-prod-down 12.4%

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

   7. Applied egg-rr 12.4%

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

   8. Taylor expanded in k around inf 12.4%

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

   if 2.9e-74 < t < 1.19999999999999994e123

   1. Initial program 57.8%

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

      1. associate-/r* 57.8%

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

      2. associate-*l* 57.8%

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

      3. associate-*l/ 57.8%

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

      4. associate-/l* 57.8%

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

      5. +-commutative 57.8%

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

      6. unpow2 57.8%

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

      7. sqr-neg 57.8%

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

      8. distribute-frac-neg 57.8%

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

      9. distribute-frac-neg 57.8%

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

      10. unpow2 57.8%

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

      11. associate--l+ 61.2%

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

      12. metadata-eval 61.2%

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

      13. +-rgt-identity 61.2%

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

      14. unpow2 61.2%

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

      15. distribute-frac-neg 61.2%

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

      16. distribute-frac-neg 61.2%

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

      17. sqr-neg 61.2%

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

      18. unpow2 61.2%

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

   3. Simplified 61.2%

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

      1. unpow3 61.2%

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

      2. times-frac 72.3%

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

      3. times-frac 83.8%

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

      4. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

   if 1.19999999999999994e123 < t

   1. Initial program 15.6%

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

   3. Taylor expanded in t around 0 73.6%

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

      1. associate-/l* 78.2%

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

   5. Simplified 78.2%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-74} \lor \neg \left(t\_m \leq 9.2 \cdot 10^{+132}\right):\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 3e-74) (not (<= t_m 9.2e+132)))
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (pow (* k (* (sin k) (sqrt t_m))) 2.0)))
    (/
     (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ t_m (/ l (tan k)))))
     (pow (/ k t_m) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 3e-74) || !(t_m <= 9.2e+132)) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / pow((k * (sin(k) * sqrt(t_m))), 2.0));
	} else {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / pow((k / t_m), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 3d-74) .or. (.not. (t_m <= 9.2d+132))) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k * (sin(k) * sqrt(t_m))) ** 2.0d0))
    else
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((t_m <= 3e-74) || !(t_m <= 9.2e+132)) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow((k * (Math.sin(k) * Math.sqrt(t_m))), 2.0));
	} else {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * (t_m / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 3e-74) or not (t_m <= 9.2e+132):
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / math.pow((k * (math.sin(k) * math.sqrt(t_m))), 2.0))
	else:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * (t_m / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 3e-74) || !(t_m <= 9.2e+132))
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / (Float64(k * Float64(sin(k) * sqrt(t_m))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(t_m / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 3e-74) || ~((t_m <= 9.2e+132)))
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k * (sin(k) * sqrt(t_m))) ^ 2.0));
	else
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 3e-74], N[Not[LessEqual[t$95$m, 9.2e+132]], $MachinePrecision]], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000007e-74 or 9.2000000000000006e132 < t

   1. Initial program 31.8%

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

      1. associate-*l* 31.8%

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

      2. associate-/r* 31.8%

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

      3. sub-neg 31.8%

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

      4. distribute-rgt-in 27.8%

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

      5. unpow2 27.8%

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

      6. times-frac 18.2%

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

      7. sqr-neg 18.2%

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

      8. times-frac 27.8%

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

      9. unpow2 27.8%

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

      10. distribute-rgt-in 31.8%

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

      11. +-commutative 31.8%

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

      12. associate-+l+ 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around 0 74.6%

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

      1. *-commutative 74.6%

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

      2. add-sqr-sqrt 22.0%

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

      3. unpow2 22.0%

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

      4. pow-prod-down 23.8%

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

   7. Applied egg-rr 23.8%

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

   8. Taylor expanded in k around inf 23.7%

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

   if 3.00000000000000007e-74 < t < 9.2000000000000006e132

   1. Initial program 53.7%

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

      1. associate-/r* 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*l/ 53.7%

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

      4. associate-/l* 53.7%

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

      5. +-commutative 53.7%

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

      6. unpow2 53.7%

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

      7. sqr-neg 53.7%

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

      8. distribute-frac-neg 53.7%

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

      9. distribute-frac-neg 53.7%

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

      10. unpow2 53.7%

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

      11. associate--l+ 60.4%

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

      12. metadata-eval 60.4%

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

      13. +-rgt-identity 60.4%

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

      14. unpow2 60.4%

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

      15. distribute-frac-neg 60.4%

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

      16. distribute-frac-neg 60.4%

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

      17. sqr-neg 60.4%

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

      18. unpow2 60.4%

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

   3. Simplified 60.4%

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

      1. unpow3 60.4%

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

      2. times-frac 70.7%

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

      3. times-frac 85.0%

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

      4. pow2 85.0%

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

   6. Applied egg-rr 85.0%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-74} \lor \neg \left(t \leq 9.2 \cdot 10^{+132}\right):\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.15 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\tan k \cdot \frac{{t\_m}^{2}}{\ell}\right)}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 3.15e-74)
      (* 2.0 (/ t_2 (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
      (if (<= t_m 5.5e+132)
        (/
         (/ 2.0 (* (* (sin k) (/ t_m l)) (* (tan k) (/ (pow t_m 2.0) l))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow k 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 3.15e-74) {
		tmp = 2.0 * (t_2 / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 5.5e+132) {
		tmp = (2.0 / ((sin(k) * (t_m / l)) * (tan(k) * (pow(t_m, 2.0) / l)))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 3.15d-74) then
        tmp = 2.0d0 * (t_2 / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 5.5d+132) then
        tmp = (2.0d0 / ((sin(k) * (t_m / l)) * (tan(k) * ((t_m ** 2.0d0) / l)))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 3.15e-74) {
		tmp = 2.0 * (t_2 / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 5.5e+132) {
		tmp = (2.0 / ((Math.sin(k) * (t_m / l)) * (Math.tan(k) * (Math.pow(t_m, 2.0) / l)))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 3.15e-74:
		tmp = 2.0 * (t_2 / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 5.5e+132:
		tmp = (2.0 / ((math.sin(k) * (t_m / l)) * (math.tan(k) * (math.pow(t_m, 2.0) / l)))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 3.15e-74)
		tmp = Float64(2.0 * Float64(t_2 / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 5.5e+132)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * Float64(t_m / l)) * Float64(tan(k) * Float64((t_m ^ 2.0) / l)))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 3.15e-74)
		tmp = 2.0 * (t_2 / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 5.5e+132)
		tmp = (2.0 / ((sin(k) * (t_m / l)) * (tan(k) * ((t_m ^ 2.0) / l)))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.15e-74], N[(2.0 * N[(t$95$2 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+132], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.15000000000000001e-74

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. unpow2 75.1%

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

      2. sin-mult 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. div-sub 72.0%

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

      2. +-inverses 72.0%

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

      3. cos-0 72.0%

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

      4. metadata-eval 72.0%

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

      5. count-2 72.0%

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

      6. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 3.15000000000000001e-74 < t < 5.5e132

   1. Initial program 53.7%

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

      1. associate-/r* 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*l/ 53.7%

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

      4. associate-/l* 53.7%

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

      5. +-commutative 53.7%

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

      6. unpow2 53.7%

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

      7. sqr-neg 53.7%

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

      8. distribute-frac-neg 53.7%

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

      9. distribute-frac-neg 53.7%

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

      10. unpow2 53.7%

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

      11. associate--l+ 60.4%

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

      12. metadata-eval 60.4%

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

      13. +-rgt-identity 60.4%

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

      14. unpow2 60.4%

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

      15. distribute-frac-neg 60.4%

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

      16. distribute-frac-neg 60.4%

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

      17. sqr-neg 60.4%

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

      18. unpow2 60.4%

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

   3. Simplified 60.4%

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

      1. cube-mult 60.4%

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

      2. times-frac 70.7%

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

      3. times-frac 85.0%

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

      4. pow2 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. associate-/r/ 84.9%

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

      2. associate-/r/ 84.9%

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

   8. Simplified 84.9%

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

   if 5.5e132 < t

   1. Initial program 16.3%

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

   3. Taylor expanded in t around 0 72.4%

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

      1. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in k around 0 70.4%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.15 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \frac{{t}^{2}}{\ell}\right)}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 3.6e-74)
      (* 2.0 (/ t_2 (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
      (if (<= t_m 1e+133)
        (/
         (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ t_m (/ l (tan k)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow k 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 3.6e-74) {
		tmp = 2.0 * (t_2 / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1e+133) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 3.6d-74) then
        tmp = 2.0d0 * (t_2 / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 1d+133) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 3.6e-74) {
		tmp = 2.0 * (t_2 / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1e+133) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * (t_m / (l / Math.tan(k))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 3.6e-74:
		tmp = 2.0 * (t_2 / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 1e+133:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * (t_m / (l / math.tan(k))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 3.6e-74)
		tmp = Float64(2.0 * Float64(t_2 / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 1e+133)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(t_m / Float64(l / tan(k))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 3.6e-74)
		tmp = 2.0 * (t_2 / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 1e+133)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * (t_m / (l / tan(k))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.6e-74], N[(2.0 * N[(t$95$2 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+133], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.6000000000000002e-74

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. unpow2 75.1%

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

      2. sin-mult 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. div-sub 72.0%

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

      2. +-inverses 72.0%

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

      3. cos-0 72.0%

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

      4. metadata-eval 72.0%

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

      5. count-2 72.0%

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

      6. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 3.6000000000000002e-74 < t < 1e133

   1. Initial program 53.7%

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

      1. associate-/r* 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*l/ 53.7%

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

      4. associate-/l* 53.7%

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

      5. +-commutative 53.7%

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

      6. unpow2 53.7%

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

      7. sqr-neg 53.7%

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

      8. distribute-frac-neg 53.7%

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

      9. distribute-frac-neg 53.7%

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

      10. unpow2 53.7%

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

      11. associate--l+ 60.4%

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

      12. metadata-eval 60.4%

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

      13. +-rgt-identity 60.4%

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

      14. unpow2 60.4%

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

      15. distribute-frac-neg 60.4%

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

      16. distribute-frac-neg 60.4%

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

      17. sqr-neg 60.4%

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

      18. unpow2 60.4%

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

   3. Simplified 60.4%

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

      1. unpow3 60.4%

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

      2. times-frac 70.7%

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

      3. times-frac 85.0%

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

      4. pow2 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 1e133 < t

   1. Initial program 16.3%

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

   3. Taylor expanded in t around 0 72.4%

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

      1. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in k around 0 70.4%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-74} \lor \neg \left(t\_m \leq 1.5 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{t\_m}^{3}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\right) \cdot {\left(\frac{k}{t\_m}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 1.8e-74) (not (<= t_m 1.5e+34)))
    (/ 2.0 (/ (pow k 2.0) (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 2.0)))))
    (*
     (* (/ 2.0 (pow t_m 3.0)) (* (/ l (tan k)) (/ l (sin k))))
     (pow (/ k t_m) -2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 1.8e-74) || !(t_m <= 1.5e+34)) {
		tmp = 2.0 / (pow(k, 2.0) / ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 2.0))));
	} else {
		tmp = ((2.0 / pow(t_m, 3.0)) * ((l / tan(k)) * (l / sin(k)))) * pow((k / t_m), -2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 1.8d-74) .or. (.not. (t_m <= 1.5d+34))) then
        tmp = 2.0d0 / ((k ** 2.0d0) / (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 2.0d0))))
    else
        tmp = ((2.0d0 / (t_m ** 3.0d0)) * ((l / tan(k)) * (l / sin(k)))) * ((k / t_m) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((t_m <= 1.8e-74) || !(t_m <= 1.5e+34)) {
		tmp = 2.0 / (Math.pow(k, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 2.0))));
	} else {
		tmp = ((2.0 / Math.pow(t_m, 3.0)) * ((l / Math.tan(k)) * (l / Math.sin(k)))) * Math.pow((k / t_m), -2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 1.8e-74) or not (t_m <= 1.5e+34):
		tmp = 2.0 / (math.pow(k, 2.0) / ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 2.0))))
	else:
		tmp = ((2.0 / math.pow(t_m, 3.0)) * ((l / math.tan(k)) * (l / math.sin(k)))) * math.pow((k / t_m), -2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 1.8e-74) || !(t_m <= 1.5e+34))
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) * Float64(Float64(l / tan(k)) * Float64(l / sin(k)))) * (Float64(k / t_m) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 1.8e-74) || ~((t_m <= 1.5e+34)))
		tmp = 2.0 / ((k ^ 2.0) / (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 2.0))));
	else
		tmp = ((2.0 / (t_m ^ 3.0)) * ((l / tan(k)) * (l / sin(k)))) * ((k / t_m) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 1.8e-74], N[Not[LessEqual[t$95$m, 1.5e+34]], $MachinePrecision]], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8000000000000001e-74 or 1.50000000000000009e34 < t

   1. Initial program 32.7%

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

   3. Taylor expanded in t around 0 75.3%

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

      1. associate-/l* 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in k around 0 68.4%

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

   if 1.8000000000000001e-74 < t < 1.50000000000000009e34

   1. Initial program 56.5%

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

      1. associate-/r* 56.5%

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

      2. associate-*l* 56.5%

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

      3. associate-*l/ 56.5%

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

      4. associate-/l* 56.5%

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

      5. +-commutative 56.5%

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

      6. unpow2 56.5%

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

      7. sqr-neg 56.5%

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

      8. distribute-frac-neg 56.5%

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

      9. distribute-frac-neg 56.5%

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

      10. unpow2 56.5%

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

      11. associate--l+ 56.8%

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

      12. metadata-eval 56.8%

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

      13. +-rgt-identity 56.8%

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

      14. unpow2 56.8%

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

      15. distribute-frac-neg 56.8%

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

      16. distribute-frac-neg 56.8%

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

      17. sqr-neg 56.8%

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

      18. unpow2 56.8%

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

   3. Simplified 56.8%

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

      1. unpow3 56.8%

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

      2. times-frac 74.9%

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

      3. times-frac 75.0%

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

      4. pow2 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. div-inv 75.0%

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

      2. frac-times 75.0%

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

      3. unpow2 75.0%

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

      4. pow3 75.0%

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

      5. pow-flip 75.0%

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

      6. metadata-eval 75.0%

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

   8. Applied egg-rr 75.0%

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

      1. associate-/r/ 74.9%

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

   10. Simplified 74.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-74} \lor \neg \left(t \leq 1.5 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{t}^{3}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\right) \cdot {\left(\frac{k}{t}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-74} \lor \neg \left(t\_m \leq 5.5 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{{t\_m}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 1.85e-74) (not (<= t_m 5.5e+65)))
    (/ 2.0 (/ (pow k 2.0) (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 2.0)))))
    (/
     (/ 2.0 (/ (* (tan k) (/ (pow t_m 3.0) l)) (/ l (sin k))))
     (pow (/ k t_m) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 1.85e-74) || !(t_m <= 5.5e+65)) {
		tmp = 2.0 / (pow(k, 2.0) / ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 2.0))));
	} else {
		tmp = (2.0 / ((tan(k) * (pow(t_m, 3.0) / l)) / (l / sin(k)))) / pow((k / t_m), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 1.85d-74) .or. (.not. (t_m <= 5.5d+65))) then
        tmp = 2.0d0 / ((k ** 2.0d0) / (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 2.0d0))))
    else
        tmp = (2.0d0 / ((tan(k) * ((t_m ** 3.0d0) / l)) / (l / sin(k)))) / ((k / t_m) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((t_m <= 1.85e-74) || !(t_m <= 5.5e+65)) {
		tmp = 2.0 / (Math.pow(k, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 2.0))));
	} else {
		tmp = (2.0 / ((Math.tan(k) * (Math.pow(t_m, 3.0) / l)) / (l / Math.sin(k)))) / Math.pow((k / t_m), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 1.85e-74) or not (t_m <= 5.5e+65):
		tmp = 2.0 / (math.pow(k, 2.0) / ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 2.0))))
	else:
		tmp = (2.0 / ((math.tan(k) * (math.pow(t_m, 3.0) / l)) / (l / math.sin(k)))) / math.pow((k / t_m), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 1.85e-74) || !(t_m <= 5.5e+65))
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / sin(k)))) / (Float64(k / t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 1.85e-74) || ~((t_m <= 5.5e+65)))
		tmp = 2.0 / ((k ^ 2.0) / (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 2.0))));
	else
		tmp = (2.0 / ((tan(k) * ((t_m ^ 3.0) / l)) / (l / sin(k)))) / ((k / t_m) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 1.85e-74], N[Not[LessEqual[t$95$m, 5.5e+65]], $MachinePrecision]], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.84999999999999997e-74 or 5.4999999999999996e65 < t

   1. Initial program 32.4%

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

   3. Taylor expanded in t around 0 75.2%

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

      1. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in k around 0 68.3%

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

   if 1.84999999999999997e-74 < t < 5.4999999999999996e65

   1. Initial program 59.0%

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

      1. associate-/r* 59.0%

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

      2. associate-*l* 59.0%

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

      3. associate-*l/ 59.0%

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

      4. associate-/l* 59.0%

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

      5. +-commutative 59.0%

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

      6. unpow2 59.0%

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

      7. sqr-neg 59.0%

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

      8. distribute-frac-neg 59.0%

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

      9. distribute-frac-neg 59.0%

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

      10. unpow2 59.0%

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

      11. associate--l+ 59.4%

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

      12. metadata-eval 59.4%

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

      13. +-rgt-identity 59.4%

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

      14. unpow2 59.4%

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

      15. distribute-frac-neg 59.4%

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

      16. distribute-frac-neg 59.4%

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

      17. sqr-neg 59.4%

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

      18. unpow2 59.4%

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

   3. Simplified 59.4%

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

      1. cube-mult 59.4%

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

      2. times-frac 76.4%

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

      3. times-frac 76.5%

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

      4. pow2 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. associate-*l/ 76.5%

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

      2. associate-*r/ 76.4%

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

      3. unpow2 76.4%

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

      4. cube-mult 76.6%

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

      5. associate-/r/ 76.5%

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

   8. Simplified 76.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-74} \lor \neg \left(t \leq 5.5 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.55 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{t\_2}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{{t\_m}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{t\_2}{t\_m \cdot {k}^{2}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 2.55e-74)
      (* 2.0 (/ t_2 (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
      (if (<= t_m 5.5e+65)
        (/
         (/ 2.0 (/ (* (tan k) (/ (pow t_m 3.0) l)) (/ l (sin k))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ t_2 (* t_m (pow k 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * cos(k);
	double tmp;
	if (t_m <= 2.55e-74) {
		tmp = 2.0 * (t_2 / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 5.5e+65) {
		tmp = (2.0 / ((tan(k) * (pow(t_m, 3.0) / l)) / (l / sin(k)))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (t_2 / (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) * cos(k)
    if (t_m <= 2.55d-74) then
        tmp = 2.0d0 * (t_2 / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 5.5d+65) then
        tmp = (2.0d0 / ((tan(k) * ((t_m ** 3.0d0) / l)) / (l / sin(k)))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / (t_2 / (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(l, 2.0) * Math.cos(k);
	double tmp;
	if (t_m <= 2.55e-74) {
		tmp = 2.0 * (t_2 / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 5.5e+65) {
		tmp = (2.0 / ((Math.tan(k) * (Math.pow(t_m, 3.0) / l)) / (l / Math.sin(k)))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (t_2 / (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) * math.cos(k)
	tmp = 0
	if t_m <= 2.55e-74:
		tmp = 2.0 * (t_2 / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 5.5e+65:
		tmp = (2.0 / ((math.tan(k) * (math.pow(t_m, 3.0) / l)) / (l / math.sin(k)))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (t_2 / (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * cos(k))
	tmp = 0.0
	if (t_m <= 2.55e-74)
		tmp = Float64(2.0 * Float64(t_2 / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 5.5e+65)
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / sin(k)))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64(t_2 / Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) * cos(k);
	tmp = 0.0;
	if (t_m <= 2.55e-74)
		tmp = 2.0 * (t_2 / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 5.5e+65)
		tmp = (2.0 / ((tan(k) * ((t_m ^ 3.0) / l)) / (l / sin(k)))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / (t_2 / (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.55e-74], N[(2.0 * N[(t$95$2 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+65], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.5499999999999998e-74

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. unpow2 75.1%

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

      2. sin-mult 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. div-sub 72.0%

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

      2. +-inverses 72.0%

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

      3. cos-0 72.0%

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

      4. metadata-eval 72.0%

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

      5. count-2 72.0%

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

      6. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 2.5499999999999998e-74 < t < 5.4999999999999996e65

   1. Initial program 59.0%

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

      1. associate-/r* 59.0%

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

      2. associate-*l* 59.0%

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

      3. associate-*l/ 59.0%

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

      4. associate-/l* 59.0%

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

      5. +-commutative 59.0%

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

      6. unpow2 59.0%

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

      7. sqr-neg 59.0%

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

      8. distribute-frac-neg 59.0%

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

      9. distribute-frac-neg 59.0%

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

      10. unpow2 59.0%

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

      11. associate--l+ 59.4%

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

      12. metadata-eval 59.4%

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

      13. +-rgt-identity 59.4%

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

      14. unpow2 59.4%

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

      15. distribute-frac-neg 59.4%

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

      16. distribute-frac-neg 59.4%

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

      17. sqr-neg 59.4%

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

      18. unpow2 59.4%

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

   3. Simplified 59.4%

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

      1. cube-mult 59.4%

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

      2. times-frac 76.4%

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

      3. times-frac 76.5%

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

      4. pow2 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. associate-*l/ 76.5%

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

      2. associate-*r/ 76.4%

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

      3. unpow2 76.4%

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

      4. cube-mult 76.6%

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

      5. associate-/r/ 76.5%

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

   8. Simplified 76.5%

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

   if 5.4999999999999996e65 < t

   1. Initial program 22.2%

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

   3. Taylor expanded in t around 0 75.8%

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

      1. associate-/l* 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in k around 0 74.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t\_2}\right)\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t\_m}^{2}}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t\_2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (pow k 2.0))))
   (*
    t_s
    (if (<= t_m 1.1e-128)
      (* 2.0 (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t_2)))
      (if (<= t_m 1.45e+45)
        (/
         (/ 2.0 (* (/ t_m (/ l (tan k))) (/ k (/ l (pow t_m 2.0)))))
         (pow (/ k t_m) 2.0))
        (/ 2.0 (/ (pow k 2.0) (/ (pow l 2.0) t_2))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * pow(k, 2.0);
	double tmp;
	if (t_m <= 1.1e-128) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t_2));
	} else if (t_m <= 1.45e+45) {
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / pow(t_m, 2.0))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (pow(l, 2.0) / t_2));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k ** 2.0d0)
    if (t_m <= 1.1d-128) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / t_2))
    else if (t_m <= 1.45d+45) then
        tmp = (2.0d0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ** 2.0d0))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / ((l ** 2.0d0) / t_2))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = t_m * Math.pow(k, 2.0);
	double tmp;
	if (t_m <= 1.1e-128) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t_2));
	} else if (t_m <= 1.45e+45) {
		tmp = (2.0 / ((t_m / (l / Math.tan(k))) * (k / (l / Math.pow(t_m, 2.0))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (Math.pow(l, 2.0) / t_2));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.pow(k, 2.0)
	tmp = 0
	if t_m <= 1.1e-128:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / t_2))
	elif t_m <= 1.45e+45:
		tmp = (2.0 / ((t_m / (l / math.tan(k))) * (k / (l / math.pow(t_m, 2.0))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (math.pow(l, 2.0) / t_2))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.1e-128)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t_2)));
	elseif (t_m <= 1.45e+45)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / Float64(l / tan(k))) * Float64(k / Float64(l / (t_m ^ 2.0))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64((l ^ 2.0) / t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (k ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.1e-128)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / t_2));
	elseif (t_m <= 1.45e+45)
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ^ 2.0))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / ((l ^ 2.0) / t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-128], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e+45], N[(N[(2.0 / N[(N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.10000000000000005e-128

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-/r* 35.6%

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

      3. sub-neg 35.6%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.2%

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

      7. sqr-neg 20.2%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.6%

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

      11. +-commutative 35.6%

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

      12. associate-+l+ 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in t around 0 76.5%

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

   6. Taylor expanded in k around 0 66.2%

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

      1. times-frac 68.3%

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

      2. *-commutative 68.3%

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

   8. Applied egg-rr 68.3%

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

   if 1.10000000000000005e-128 < t < 1.4499999999999999e45

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 42.8%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 46.8%

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

      12. metadata-eval 46.8%

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

      13. +-rgt-identity 46.8%

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

      14. unpow2 46.8%

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

      15. distribute-frac-neg 46.8%

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

      16. distribute-frac-neg 46.8%

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

      17. sqr-neg 46.8%

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

      18. unpow2 46.8%

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

   3. Simplified 46.8%

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

      1. unpow3 46.8%

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

      2. times-frac 58.1%

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

      3. times-frac 61.8%

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

      4. pow2 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in k around 0 58.7%

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

      1. associate-/l* 62.4%

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

   9. Simplified 62.4%

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

   if 1.4499999999999999e45 < t

   1. Initial program 23.6%

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

   3. Taylor expanded in t around 0 76.2%

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

      1. associate-/l* 80.1%

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

   5. Simplified 80.1%

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

   6. Taylor expanded in k around 0 74.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t}^{2}}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(k \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t\_m}^{2}}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-125)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (pow (* k (* k (sqrt t_m))) 2.0)))
    (if (<= t_m 3.2e+40)
      (/
       (/ 2.0 (* (/ t_m (/ l (tan k))) (/ k (/ l (pow t_m 2.0)))))
       (pow (/ k t_m) 2.0))
      (/ 2.0 (/ (pow k 2.0) (/ (pow l 2.0) (* t_m (pow k 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-125) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / pow((k * (k * sqrt(t_m))), 2.0));
	} else if (t_m <= 3.2e+40) {
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / pow(t_m, 2.0))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (pow(l, 2.0) / (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d-125) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k * (k * sqrt(t_m))) ** 2.0d0))
    else if (t_m <= 3.2d+40) then
        tmp = (2.0d0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ** 2.0d0))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / ((l ** 2.0d0) / (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 8e-125) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow((k * (k * Math.sqrt(t_m))), 2.0));
	} else if (t_m <= 3.2e+40) {
		tmp = (2.0 / ((t_m / (l / Math.tan(k))) * (k / (l / Math.pow(t_m, 2.0))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e-125:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / math.pow((k * (k * math.sqrt(t_m))), 2.0))
	elif t_m <= 3.2e+40:
		tmp = (2.0 / ((t_m / (l / math.tan(k))) * (k / (l / math.pow(t_m, 2.0))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-125)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / (Float64(k * Float64(k * sqrt(t_m))) ^ 2.0)));
	elseif (t_m <= 3.2e+40)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / Float64(l / tan(k))) * Float64(k / Float64(l / (t_m ^ 2.0))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-125)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k * (k * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 3.2e+40)
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ^ 2.0))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / ((l ^ 2.0) / (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-125], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+40], N[(N[(2.0 / N[(N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.0000000000000001e-125

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-/r* 35.6%

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

      3. sub-neg 35.6%

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

      4. distribute-rgt-in 30.5%

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

      5. unpow2 30.5%

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

      6. times-frac 20.2%

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

      7. sqr-neg 20.2%

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

      8. times-frac 30.5%

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

      9. unpow2 30.5%

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

      10. distribute-rgt-in 35.6%

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

      11. +-commutative 35.6%

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

      12. associate-+l+ 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in t around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. add-sqr-sqrt 8.0%

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

      3. unpow2 8.0%

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

      4. pow-prod-down 9.7%

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

   7. Applied egg-rr 9.7%

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

   8. Taylor expanded in k around 0 5.9%

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

   if 8.0000000000000001e-125 < t < 3.19999999999999981e40

   1. Initial program 46.6%

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

      1. associate-/r* 46.6%

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

      2. associate-*l* 46.6%

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

      3. associate-*l/ 42.8%

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

      4. associate-/l* 46.6%

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

      5. +-commutative 46.6%

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

      6. unpow2 46.6%

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

      7. sqr-neg 46.6%

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

      8. distribute-frac-neg 46.6%

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

      9. distribute-frac-neg 46.6%

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

      10. unpow2 46.6%

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

      11. associate--l+ 46.8%

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

      12. metadata-eval 46.8%

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

      13. +-rgt-identity 46.8%

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

      14. unpow2 46.8%

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

      15. distribute-frac-neg 46.8%

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

      16. distribute-frac-neg 46.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 46.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 46.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 46.8%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 58.1%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 61.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 61.8%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 61.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in k around 0 58.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 62.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 62.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   if 3.19999999999999981e40 < t

   1. Initial program 23.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 76.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

   5. Simplified 80.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

   6. Taylor expanded in k around 0 74.6%

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(k \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t}^{2}}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{2}}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (/ 2.0 (/ (pow k 2.0) (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow(k, 2.0) / ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 2.0)))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((k ** 2.0d0) / (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 2.0d0)))))
end function

Java

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) {
	return t_s * (2.0 / (Math.pow(k, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 2.0)))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow(k, 2.0) / ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 2.0)))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((k ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 2.0))))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((k ^ 2.0) / (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 2.0)))));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0 74.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-/l* 74.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

5. Simplified 74.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

6. Taylor expanded in k around 0 67.7%

   \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot t}}}} \]

7. Final simplification 67.7%

   \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{2}}}} \]
8. Add Preprocessing

Alternative 17: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-141}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t\_m}^{2}}}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-141)
    (* 2.0 (* (/ (pow l 2.0) (pow k 4.0)) (/ (cos k) t_m)))
    (if (<= t_m 6e+41)
      (/
       (/ 2.0 (* (/ t_m (/ l (tan k))) (/ k (/ l (pow t_m 2.0)))))
       (pow (/ k t_m) 2.0))
      (/ 2.0 (/ (pow k 2.0) (/ (pow l 2.0) (* t_m (pow k 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-141) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) * (cos(k) / t_m));
	} else if (t_m <= 6e+41) {
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / pow(t_m, 2.0))))) / pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) / (pow(l, 2.0) / (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3d-141) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) * (cos(k) / t_m))
    else if (t_m <= 6d+41) then
        tmp = (2.0d0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ** 2.0d0))))) / ((k / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k ** 2.0d0) / ((l ** 2.0d0) / (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 3e-141) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (Math.cos(k) / t_m));
	} else if (t_m <= 6e+41) {
		tmp = (2.0 / ((t_m / (l / Math.tan(k))) * (k / (l / Math.pow(t_m, 2.0))))) / Math.pow((k / t_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) / (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3e-141:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) * (math.cos(k) / t_m))
	elif t_m <= 6e+41:
		tmp = (2.0 / ((t_m / (l / math.tan(k))) * (k / (l / math.pow(t_m, 2.0))))) / math.pow((k / t_m), 2.0)
	else:
		tmp = 2.0 / (math.pow(k, 2.0) / (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-141)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(cos(k) / t_m)));
	elseif (t_m <= 6e+41)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / Float64(l / tan(k))) * Float64(k / Float64(l / (t_m ^ 2.0))))) / (Float64(k / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3e-141)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) * (cos(k) / t_m));
	elseif (t_m <= 6e+41)
		tmp = (2.0 / ((t_m / (l / tan(k))) * (k / (l / (t_m ^ 2.0))))) / ((k / t_m) ^ 2.0);
	else
		tmp = 2.0 / ((k ^ 2.0) / ((l ^ 2.0) / (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-141], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+41], N[(N[(2.0 / N[(N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.99999999999999983e-141

   1. Initial program 35.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 30.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 30.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 20.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 20.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 30.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 30.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 41.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 66.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. Taylor expanded in l around 0 64.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. times-frac 64.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right)} \]

   9. Simplified 64.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right)} \]

   if 2.99999999999999983e-141 < t < 5.9999999999999997e41

   1. Initial program 44.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 44.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 44.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 41.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 44.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 44.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 45.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 45.1%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 56.0%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 59.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 59.5%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 59.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in k around 0 60.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 63.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 63.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   if 5.9999999999999997e41 < t

   1. Initial program 23.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 76.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

   5. Simplified 80.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

   6. Taylor expanded in k around 0 74.6%

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-141}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \frac{k}{\frac{\ell}{{t}^{2}}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (pow k 2.0) (/ (pow l 2.0) (* t_m (pow k 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow(k, 2.0) / (pow(l, 2.0) / (t_m * pow(k, 2.0)))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((k ** 2.0d0) / ((l ** 2.0d0) / (t_m * (k ** 2.0d0)))))
end function

Java

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) {
	return t_s * (2.0 / (Math.pow(k, 2.0) / (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0)))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow(k, 2.0) / (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0)))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((k ^ 2.0) / Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((k ^ 2.0) / ((l ^ 2.0) / (t_m * (k ^ 2.0)))));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0 74.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-/l* 74.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

5. Simplified 74.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

6. Taylor expanded in k around 0 65.2%

   \[\leadsto \frac{2}{\frac{{k}^{2}}{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}}} \]

7. Final simplification 65.2%

   \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}} \]
8. Add Preprocessing

Alternative 19: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t\_m}\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (/ (pow l 2.0) (pow k 4.0)) (/ (cos k) t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) / pow(k, 4.0)) * (cos(k) / t_m)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) * (cos(k) / t_m)))
end function

Java

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) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (Math.cos(k) / t_m)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) * (math.cos(k) / t_m)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(cos(k) / t_m))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) / (k ^ 4.0)) * (cos(k) / t_m)));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 34.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 34.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 34.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 43.1%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 43.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

6. Taylor expanded in k around 0 66.1%

   \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

7. Taylor expanded in l around 0 65.2%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. times-frac 64.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right)} \]

9. Simplified 64.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right)} \]

10. Final simplification 64.5%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\cos k}{t}\right) \]
11. Add Preprocessing

Alternative 20: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0))))
end function

Java

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) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0))));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 34.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 34.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 34.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 43.1%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 43.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

6. Taylor expanded in k around 0 65.2%

   \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

7. Final simplification 65.2%

   \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}} \]
8. Add Preprocessing

Alternative 21: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
end function

Java

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) {
	return t_s * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 34.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 34.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 34.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 22.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 30.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 34.2%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 43.1%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 43.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 63.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

6. Final simplification 63.8%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 22: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{t\_m \cdot {k}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (* t_m (pow k 4.0)) (pow l 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((t_m * pow(k, 4.0)) / pow(l, 2.0)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((t_m * (k ** 4.0d0)) / (l ** 2.0d0)))
end function

Java

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) {
	return t_s * (2.0 / ((t_m * Math.pow(k, 4.0)) / Math.pow(l, 2.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((t_m * math.pow(k, 4.0)) / math.pow(l, 2.0)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(t_m * (k ^ 4.0)) / (l ^ 2.0))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((t_m * (k ^ 4.0)) / (l ^ 2.0)));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 / N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0 74.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-/l* 74.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]

5. Simplified 74.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 28.2%

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\sqrt{t \cdot {\sin k}^{2}} \cdot \sqrt{t \cdot {\sin k}^{2}}}}}} \]

   2. pow2 28.2%

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\left(\sqrt{t \cdot {\sin k}^{2}}\right)}^{2}}}}} \]

7. Applied egg-rr 28.2%

   \[\leadsto \frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\left(\sqrt{t \cdot {\sin k}^{2}}\right)}^{2}}}}} \]

8. Taylor expanded in k around 0 63.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

9. Final simplification 63.8%

   \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))