Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 83.0%

Time: 18.8s

Alternatives: 19

Speedup: 1.9×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}}{t\_2} \cdot \frac{\ell}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m)))))
   (*
    t_s
    (if (<= t_m 1.3e-89)
      (*
       2.0
       (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 3.5e+99)
        (*
         (/ (/ (* 2.0 l) (* (pow t_m 3.0) (* (sin k) (tan k)))) t_2)
         (/ l t_2))
        (/
         2.0
         (*
          (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
          (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ 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) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.3e-89) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 3.5e+99) {
		tmp = (((2.0 * l) / (pow(t_m, 3.0) * (sin(k) * tan(k)))) / t_2) * (l / t_2);
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))));
	}
	return t_s * tmp;
}

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.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.3e-89) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 3.5e+99) {
		tmp = (((2.0 * l) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) / t_2) * (l / t_2);
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))));
	}
	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 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	tmp = 0.0
	if (t_m <= 1.3e-89)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 3.5e+99)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) / t_2) * Float64(l / t_2));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))))));
	end
	return Float64(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[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-89], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+99], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.2999999999999999e-89

   1. Initial program 49.1%

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

      1. associate-/r* 49.1%

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

      2. +-commutative 49.1%

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

      3. unpow2 49.1%

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

      4. sqr-neg 49.1%

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

      5. distribute-frac-neg2 49.1%

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

      6. distribute-frac-neg2 49.1%

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

      7. unpow2 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-*l* 44.3%

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

      10. associate-*l/ 44.4%

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

      11. associate-/r/ 44.2%

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

      12. +-commutative 44.2%

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

      13. associate-+r+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 63.3%

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

   if 1.2999999999999999e-89 < t < 3.4999999999999998e99

   1. Initial program 85.5%

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

      1. associate-/r* 86.0%

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

      2. +-commutative 86.0%

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

      3. unpow2 86.0%

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

      4. sqr-neg 86.0%

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

      5. distribute-frac-neg2 86.0%

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

      6. distribute-frac-neg2 86.0%

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

      7. unpow2 86.0%

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

      8. +-commutative 86.0%

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

      9. associate-*l* 83.1%

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

      10. associate-*l/ 83.1%

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

      11. associate-/r/ 83.3%

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

      12. +-commutative 83.3%

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

      13. associate-+r+ 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*r* 86.1%

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

      2. add-sqr-sqrt 85.9%

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

      3. times-frac 85.9%

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

      4. metadata-eval 85.9%

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

      5. associate-+r+ 85.9%

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

      6. add-sqr-sqrt 85.9%

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

      7. hypot-1-def 85.9%

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

      8. unpow2 85.9%

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

      9. hypot-1-def 85.9%

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

      10. metadata-eval 85.9%

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

   6. Applied egg-rr 94.0%

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

      1. associate-*l/ 94.1%

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

   8. Simplified 94.1%

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

   if 3.4999999999999998e99 < t

   1. Initial program 78.6%

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

   3. Simplified 78.6%

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

      1. unpow2 78.6%

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

   6. Applied egg-rr 78.6%

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

      1. add-cube-cbrt 78.6%

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

      2. pow3 78.6%

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

      3. *-commutative 78.6%

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

      4. cbrt-prod 78.6%

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

      5. cbrt-div 78.6%

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

      6. rem-cbrt-cube 89.2%

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

      7. cbrt-prod 99.4%

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

      8. pow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-66}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.1e-66)
      (*
       2.0
       (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 2.6e+97)
        (*
         (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
         (/ l (+ 2.0 t_2)))
        (/
         2.0
         (*
          (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0)
          (* (tan k) (+ 1.0 (+ 1.0 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 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.1e-66) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 2.6e+97) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0) * (tan(k) * (1.0 + (1.0 + t_2))));
	}
	return t_s * tmp;
}

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((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.1e-66) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 2.6e+97) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (Math.tan(k) * (1.0 + (1.0 + 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(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.1e-66)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 2.6e+97)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + t_2)));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	end
	return Float64(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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-66], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.6e+97], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1000000000000001e-66

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

   if 1.1000000000000001e-66 < t < 2.6e97

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 2.6e97 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. *-commutative 77.2%

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

      4. cbrt-prod 77.2%

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

      5. cbrt-div 77.2%

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

      6. rem-cbrt-cube 87.0%

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

      7. cbrt-prod 96.4%

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

      8. pow2 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

      1. associate-*l/ 96.5%

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

   10. Applied egg-rr 96.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-66}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.7 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.7e-67)
    (*
     2.0
     (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 1.1e+96)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
        (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ 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) {
	double tmp;
	if (t_m <= 5.7e-67) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 1.1e+96) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))));
	}
	return t_s * tmp;
}

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 <= 5.7e-67) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 1.1e+96) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))));
	}
	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 <= 5.7e-67)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 1.1e+96)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))))));
	end
	return Float64(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, 5.7e-67], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+96], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.7000000000000002e-67

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

   if 5.7000000000000002e-67 < t < 1.0999999999999999e96

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 1.0999999999999999e96 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. unpow2 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. *-commutative 77.2%

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

      4. cbrt-prod 77.2%

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

      5. cbrt-div 77.2%

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

      6. rem-cbrt-cube 87.0%

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

      7. cbrt-prod 96.4%

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

      8. pow2 96.4%

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

   8. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

   10. Simplified 96.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-67)
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (sin k)) 2.0))))
    (if (<= t_m 4e+95)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
        (* 2.0 k)))))))

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 <= 8.5e-67) {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * sin(k)), 2.0)));
	} else if (t_m <= 4e+95) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

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 <= 8.5e-67) {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t_m <= 4e+95) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	}
	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 <= 8.5e-67)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))));
	elseif (t_m <= 4e+95)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(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, 8.5e-67], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+95], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.49999999999999993e-67

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. times-frac 63.2%

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

      3. associate-/r* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in k around inf 64.1%

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

      1. associate-/l* 63.5%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*l* 62.4%

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

      5. *-commutative 62.4%

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

      6. unpow2 62.4%

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

      7. unpow2 62.4%

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

      8. swap-sqr 62.4%

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

      9. unpow2 62.4%

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

   10. Simplified 62.4%

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

   if 8.49999999999999993e-67 < t < 4.00000000000000008e95

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 4.00000000000000008e95 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. *-commutative 77.2%

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

      4. cbrt-prod 77.2%

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

      5. cbrt-div 77.2%

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

      6. rem-cbrt-cube 87.0%

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

      7. cbrt-prod 96.4%

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

      8. pow2 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in k around 0 89.6%

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

       1. *-commutative 80.1%

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

   11. Simplified 89.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-67)
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (sin k)) 2.0))))
    (if (<= t_m 2.5e+97)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0)
        (* 2.0 k)))))))

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 <= 6.8e-67) {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * sin(k)), 2.0)));
	} else if (t_m <= 2.5e+97) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

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 <= 6.8e-67) {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t_m <= 2.5e+97) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * k));
	}
	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 <= 6.8e-67)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))));
	elseif (t_m <= 2.5e+97)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(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, 6.8e-67], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+97], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.8000000000000002e-67

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. times-frac 63.2%

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

      3. associate-/r* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in k around inf 64.1%

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

      1. associate-/l* 63.5%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*l* 62.4%

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

      5. *-commutative 62.4%

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

      6. unpow2 62.4%

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

      7. unpow2 62.4%

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

      8. swap-sqr 62.4%

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

      9. unpow2 62.4%

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

   10. Simplified 62.4%

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

   if 6.8000000000000002e-67 < t < 2.49999999999999999e97

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 2.49999999999999999e97 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. *-commutative 77.2%

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

      4. cbrt-prod 77.2%

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

      5. cbrt-div 77.2%

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

      6. rem-cbrt-cube 87.0%

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

      7. cbrt-prod 96.4%

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

      8. pow2 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

      1. associate-*l/ 96.5%

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

   10. Applied egg-rr 96.5%

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

   11. Taylor expanded in k around 0 89.7%

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

       1. *-commutative 80.1%

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

   13. Simplified 89.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 3.05 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.4e-65)
    (*
     2.0
     (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 3.05e+97)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0)
        (* 2.0 k)))))))

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 <= 1.4e-65) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 3.05e+97) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

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 <= 1.4e-65) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 3.05e+97) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * k));
	}
	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 <= 1.4e-65)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 3.05e+97)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(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, 1.4e-65], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.05e+97], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.4e-65

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

   if 1.4e-65 < t < 3.05e97

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 3.05e97 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. *-commutative 77.2%

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

      4. cbrt-prod 77.2%

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

      5. cbrt-div 77.2%

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

      6. rem-cbrt-cube 87.0%

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

      7. cbrt-prod 96.4%

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

      8. pow2 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

      1. associate-*l/ 96.5%

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

   10. Applied egg-rr 96.5%

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

   11. Taylor expanded in k around 0 89.7%

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

       1. *-commutative 80.1%

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

   13. Simplified 89.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-67)
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (sin k)) 2.0))))
    (if (<= t_m 2.9e+97)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m)))))
        (* (sin k) (pow (/ (pow t_m 1.5) 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) {
	double tmp;
	if (t_m <= 5.5e-67) {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * sin(k)), 2.0)));
	} else if (t_m <= 2.9e+97) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * pow((pow(t_m, 1.5) / l), 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 <= 5.5d-67) then
        tmp = (2.0d0 * (l ** 2.0d0)) * (cos(k) / (t_m * ((k * sin(k)) ** 2.0d0)))
    else if (t_m <= 2.9d+97) then
        tmp = ((2.0d0 * (t_m ** (-3.0d0))) * ((l / tan(k)) / sin(k))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ** 1.5d0) / l) ** 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 <= 5.5e-67) {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t_m <= 2.9e+97) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 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 <= 5.5e-67:
		tmp = (2.0 * math.pow(l, 2.0)) * (math.cos(k) / (t_m * math.pow((k * math.sin(k)), 2.0)))
	elif t_m <= 2.9e+97:
		tmp = ((2.0 * math.pow(t_m, -3.0)) * ((l / math.tan(k)) / math.sin(k))) * (l / (2.0 + math.pow((k / t_m), 2.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 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 <= 5.5e-67)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))));
	elseif (t_m <= 2.9e+97)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 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 <= 5.5e-67)
		tmp = (2.0 * (l ^ 2.0)) * (cos(k) / (t_m * ((k * sin(k)) ^ 2.0)));
	elseif (t_m <= 2.9e+97)
		tmp = ((2.0 * (t_m ^ -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ^ 1.5) / l) ^ 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, 5.5e-67], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+97], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5000000000000003e-67

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. times-frac 63.2%

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

      3. associate-/r* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in k around inf 64.1%

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

      1. associate-/l* 63.5%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*l* 62.4%

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

      5. *-commutative 62.4%

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

      6. unpow2 62.4%

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

      7. unpow2 62.4%

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

      8. swap-sqr 62.4%

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

      9. unpow2 62.4%

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

   10. Simplified 62.4%

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

   if 5.5000000000000003e-67 < t < 2.89999999999999987e97

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 2.89999999999999987e97 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. unpow2 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. add-sqr-sqrt 77.2%

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

      2. pow2 77.2%

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

      3. sqrt-div 77.2%

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

      4. sqrt-pow1 84.6%

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

      5. metadata-eval 84.6%

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

      6. sqrt-prod 64.3%

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

      7. add-sqr-sqrt 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-66}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(2 \cdot {t\_m}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-66)
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (sin k)) 2.0))))
    (if (<= t_m 1.35e+93)
      (*
       (* (* 2.0 (pow t_m -3.0)) (/ (/ l (tan k)) (sin k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m)))))
        (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))))))

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 <= 1e-66) {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * sin(k)), 2.0)));
	} else if (t_m <= 1.35e+93) {
		tmp = ((2.0 * pow(t_m, -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 1d-66) then
        tmp = (2.0d0 * (l ** 2.0d0)) * (cos(k) / (t_m * ((k * sin(k)) ** 2.0d0)))
    else if (t_m <= 1.35d+93) then
        tmp = ((2.0d0 * (t_m ** (-3.0d0))) * ((l / tan(k)) / sin(k))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    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 <= 1e-66) {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t_m <= 1.35e+93) {
		tmp = ((2.0 * Math.pow(t_m, -3.0)) * ((l / Math.tan(k)) / Math.sin(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 1e-66:
		tmp = (2.0 * math.pow(l, 2.0)) * (math.cos(k) / (t_m * math.pow((k * math.sin(k)), 2.0)))
	elif t_m <= 1.35e+93:
		tmp = ((2.0 * math.pow(t_m, -3.0)) * ((l / math.tan(k)) / math.sin(k))) * (l / (2.0 + math.pow((k / t_m), 2.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	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 <= 1e-66)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))));
	elseif (t_m <= 1.35e+93)
		tmp = Float64(Float64(Float64(2.0 * (t_m ^ -3.0)) * Float64(Float64(l / tan(k)) / sin(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	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 <= 1e-66)
		tmp = (2.0 * (l ^ 2.0)) * (cos(k) / (t_m * ((k * sin(k)) ^ 2.0)));
	elseif (t_m <= 1.35e+93)
		tmp = ((2.0 * (t_m ^ -3.0)) * ((l / tan(k)) / sin(k))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	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, 1e-66], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+93], N[(N[(N[(2.0 * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.9999999999999998e-67

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.6%

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

      11. associate-/r/ 45.4%

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

      12. +-commutative 45.4%

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

      13. associate-+r+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. times-frac 63.2%

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

      3. associate-/r* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in k around inf 64.1%

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

      1. associate-/l* 63.5%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*l* 62.4%

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

      5. *-commutative 62.4%

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

      6. unpow2 62.4%

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

      7. unpow2 62.4%

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

      8. swap-sqr 62.4%

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

      9. unpow2 62.4%

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

   10. Simplified 62.4%

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

   if 9.9999999999999998e-67 < t < 1.35e93

   1. Initial program 86.3%

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

      1. associate-/r* 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. distribute-frac-neg2 86.2%

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

      6. distribute-frac-neg2 86.2%

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

      7. unpow2 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-*l* 82.6%

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

      10. associate-*l/ 82.6%

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

      11. associate-/r/ 82.8%

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

      12. +-commutative 82.8%

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

      13. associate-+r+ 82.8%

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

   3. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. associate-*r* 86.1%

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

      3. associate-*l* 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.2%

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

      2. times-frac 86.1%

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

      3. associate-*r/ 86.1%

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

      4. *-rgt-identity 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

      2. div-inv 86.0%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. associate-/l* 86.1%

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

       2. *-commutative 86.1%

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

       3. associate-/r* 89.6%

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

   12. Simplified 89.6%

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

   if 1.35e93 < t

   1. Initial program 77.2%

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

   3. Simplified 77.2%

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

      1. unpow2 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow3 77.2%

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

      2. times-frac 92.2%

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

      3. pow2 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-66}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(2 \cdot {t}^{-3}\right) \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-90)
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (sin k)) 2.0))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m)))))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))))

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 <= 6.8e-90) {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * sin(k)), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 6.8d-90) then
        tmp = (2.0d0 * (l ** 2.0d0)) * (cos(k) / (t_m * ((k * sin(k)) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    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 <= 6.8e-90) {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * Math.sin(k)), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 6.8e-90:
		tmp = (2.0 * math.pow(l, 2.0)) * (math.cos(k) / (t_m * math.pow((k * math.sin(k)), 2.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	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 <= 6.8e-90)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	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 <= 6.8e-90)
		tmp = (2.0 * (l ^ 2.0)) * (cos(k) / (t_m * ((k * sin(k)) ^ 2.0)));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	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, 6.8e-90], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.79999999999999988e-90

   1. Initial program 49.1%

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

      1. associate-/r* 49.1%

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

      2. +-commutative 49.1%

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

      3. unpow2 49.1%

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

      4. sqr-neg 49.1%

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

      5. distribute-frac-neg2 49.1%

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

      6. distribute-frac-neg2 49.1%

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

      7. unpow2 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-*l* 44.3%

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

      10. associate-*l/ 44.4%

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

      11. associate-/r/ 44.2%

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

      12. +-commutative 44.2%

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

      13. associate-+r+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 63.3%

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

      1. associate-*r/ 63.3%

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

      2. times-frac 62.5%

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

      3. associate-/r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in k around inf 63.3%

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

      1. associate-/l* 62.7%

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

      2. associate-*r* 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-*l* 61.6%

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

      5. *-commutative 61.6%

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

      6. unpow2 61.6%

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

      7. unpow2 61.6%

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

      8. swap-sqr 61.6%

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

      9. unpow2 61.6%

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

   10. Simplified 61.6%

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

   if 6.79999999999999988e-90 < t

   1. Initial program 82.0%

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

   3. Simplified 81.9%

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

      1. unpow2 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. unpow3 81.9%

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

      2. times-frac 90.3%

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

      3. pow2 90.3%

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

   8. Applied egg-rr 90.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t\_m \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e-20)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (* (* 2.0 (pow l 2.0)) (/ (cos k) (* t_m (pow (* k (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 (k <= 9.2e-20) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = (2.0 * pow(l, 2.0)) * (cos(k) / (t_m * pow((k * 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 (k <= 9.2d-20) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) * (cos(k) / (t_m * ((k * 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 (k <= 9.2e-20) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) * (Math.cos(k) / (t_m * Math.pow((k * 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 k <= 9.2e-20:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) * (math.cos(k) / (t_m * math.pow((k * 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 (k <= 9.2e-20)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) * Float64(cos(k) / Float64(t_m * (Float64(k * 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 (k <= 9.2e-20)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = (2.0 * (l ^ 2.0)) * (cos(k) / (t_m * ((k * 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[k, 9.2e-20], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.1999999999999997e-20

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. *-commutative 28.9%

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

      4. sqrt-prod 11.4%

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

      5. sqrt-div 11.4%

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

      6. sqrt-pow1 13.1%

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

      7. metadata-eval 13.1%

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

      8. sqrt-prod 8.5%

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

      9. add-sqr-sqrt 14.2%

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

   6. Applied egg-rr 14.2%

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

      1. *-commutative 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in k around 0 13.3%

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

       1. *-commutative 64.8%

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

   11. Simplified 13.3%

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

   if 9.1999999999999997e-20 < k

   1. Initial program 50.0%

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

      1. associate-/r* 50.3%

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

      2. +-commutative 50.3%

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

      3. unpow2 50.3%

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

      4. sqr-neg 50.3%

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

      5. distribute-frac-neg2 50.3%

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

      6. distribute-frac-neg2 50.3%

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

      7. unpow2 50.3%

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

      8. +-commutative 50.3%

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

      9. associate-*l* 50.3%

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

      10. associate-*l/ 50.3%

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

      11. associate-/r/ 50.3%

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

      12. +-commutative 50.3%

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

      13. associate-+r+ 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. associate-*r/ 72.8%

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

      2. times-frac 70.5%

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

      3. associate-/r* 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in k around inf 72.8%

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

      1. associate-/l* 71.7%

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

      2. associate-*r* 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-*l* 71.7%

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

      5. *-commutative 71.7%

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

      6. unpow2 71.7%

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

      7. unpow2 71.7%

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

      8. swap-sqr 71.7%

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

      9. unpow2 71.7%

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

   10. Simplified 71.7%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ (pow t_m 3.0) l) l)))
   (*
    t_s
    (if (<= t_m 2.25e-90)
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))
      (if (<= t_m 4e+99)
        (/ 2.0 (* t_2 (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))))
        (/ 2.0 (* (* 2.0 k) (* (sin 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 = (pow(t_m, 3.0) / l) / l;
	double tmp;
	if (t_m <= 2.25e-90) {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	} else if (t_m <= 4e+99) {
		tmp = 2.0 / (t_2 * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(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 ** 3.0d0) / l) / l
    if (t_m <= 2.25d-90) then
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    else if (t_m <= 4d+99) then
        tmp = 2.0d0 / (t_2 * ((sin(k) * tan(k)) * (2.0d0 + ((k / t_m) / (t_m / k)))))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(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 = (Math.pow(t_m, 3.0) / l) / l;
	double tmp;
	if (t_m <= 2.25e-90) {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	} else if (t_m <= 4e+99) {
		tmp = 2.0 / (t_2 * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(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 = (math.pow(t_m, 3.0) / l) / l
	tmp = 0
	if t_m <= 2.25e-90:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	elif t_m <= 4e+99:
		tmp = 2.0 / (t_2 * ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(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(Float64((t_m ^ 3.0) / l) / l)
	tmp = 0.0
	if (t_m <= 2.25e-90)
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	elseif (t_m <= 4e+99)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(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 ^ 3.0) / l) / l;
	tmp = 0.0;
	if (t_m <= 2.25e-90)
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	elseif (t_m <= 4e+99)
		tmp = 2.0 / (t_2 * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(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[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.25e-90], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+99], N[(2.0 / N[(t$95$2 * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.25000000000000004e-90

   1. Initial program 49.1%

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

      1. associate-/r* 49.1%

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

      2. +-commutative 49.1%

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

      3. unpow2 49.1%

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

      4. sqr-neg 49.1%

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

      5. distribute-frac-neg2 49.1%

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

      6. distribute-frac-neg2 49.1%

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

      7. unpow2 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-*l* 44.3%

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

      10. associate-*l/ 44.4%

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

      11. associate-/r/ 44.2%

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

      12. +-commutative 44.2%

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

      13. associate-+r+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 63.3%

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

      1. associate-*r/ 63.3%

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

      2. times-frac 62.5%

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

      3. associate-/r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in k around 0 50.1%

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

      1. pow2 50.1%

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

      2. add-sqr-sqrt 34.2%

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

      3. sqrt-div 14.4%

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

      4. sqrt-prod 7.2%

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

      5. add-sqr-sqrt 8.5%

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

      6. *-commutative 8.5%

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

      7. sqrt-prod 8.5%

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

      8. sqrt-pow1 8.5%

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

      9. metadata-eval 8.5%

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

      10. sqrt-div 8.5%

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

      11. sqrt-prod 7.7%

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

      12. add-sqr-sqrt 15.3%

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

      13. *-commutative 15.3%

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

      14. sqrt-prod 15.3%

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

      15. sqrt-pow1 15.4%

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

      16. metadata-eval 15.4%

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

   10. Applied egg-rr 15.4%

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

       1. unpow2 15.4%

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

       2. *-commutative 15.4%

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

   12. Simplified 15.4%

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

   if 2.25000000000000004e-90 < t < 3.9999999999999999e99

   1. Initial program 85.5%

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

   3. Simplified 82.8%

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

      1. unpow2 82.8%

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

      2. clear-num 82.8%

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

      3. un-div-inv 82.8%

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

   6. Applied egg-rr 82.8%

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

   if 3.9999999999999999e99 < t

   1. Initial program 78.6%

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

   3. Simplified 78.6%

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

      1. associate-/r* 81.7%

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

      2. div-inv 81.7%

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

      3. add-cube-cbrt 81.7%

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

      4. associate-*l* 81.7%

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

      5. pow2 81.7%

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

      6. cbrt-div 81.7%

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

      7. rem-cbrt-cube 81.7%

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

      8. cbrt-div 81.7%

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

      9. rem-cbrt-cube 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. associate-*r* 92.0%

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

      2. pow-plus 92.0%

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

      3. metadata-eval 92.0%

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

      4. add-cbrt-cube 81.7%

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

      5. unpow3 81.7%

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

      6. cbrt-div 81.7%

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

      7. pow3 81.7%

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

      8. add-cube-cbrt 81.7%

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

      9. div-inv 81.7%

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

   8. Applied egg-rr 81.7%

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

   9. Taylor expanded in k around 0 81.7%

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

       1. *-commutative 81.7%

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

   11. Simplified 81.7%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-126)
    (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m)))))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))))

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-126) {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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-126) then
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    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-126) {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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-126:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	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-126)
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	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-126)
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	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, 1.85e-126], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e-126

   1. Initial program 50.4%

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

      1. associate-/r* 50.4%

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

      2. +-commutative 50.4%

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

      3. unpow2 50.4%

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

      4. sqr-neg 50.4%

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

      5. distribute-frac-neg2 50.4%

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

      6. distribute-frac-neg2 50.4%

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

      7. unpow2 50.4%

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

      8. +-commutative 50.4%

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

      9. associate-*l* 45.4%

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

      10. associate-*l/ 45.5%

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

      11. associate-/r/ 45.3%

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

      12. +-commutative 45.3%

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

      13. associate-+r+ 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 63.0%

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

      1. associate-*r/ 63.0%

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

      2. times-frac 62.1%

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

      3. associate-/r* 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in k around 0 50.9%

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

      1. pow2 50.9%

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

      2. add-sqr-sqrt 34.4%

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

      3. sqrt-div 13.8%

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

      4. sqrt-prod 6.9%

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

      5. add-sqr-sqrt 8.1%

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

      6. *-commutative 8.1%

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

      7. sqrt-prod 8.1%

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

      8. sqrt-pow1 8.1%

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

      9. metadata-eval 8.1%

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

      10. sqrt-div 8.1%

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

      11. sqrt-prod 7.5%

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

      12. add-sqr-sqrt 14.7%

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

      13. *-commutative 14.7%

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

      14. sqrt-prod 14.7%

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

      15. sqrt-pow1 14.8%

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

      16. metadata-eval 14.8%

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

   10. Applied egg-rr 14.8%

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

       1. unpow2 14.8%

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

       2. *-commutative 14.8%

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

   12. Simplified 14.8%

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

   if 1.85e-126 < t

   1. Initial program 76.2%

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

   3. Simplified 76.1%

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

      1. unpow2 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. unpow3 76.1%

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

      2. times-frac 86.2%

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

      3. pow2 86.2%

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

   8. Applied egg-rr 86.2%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.3e-90)
    (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m)))))
      (* (sin k) (/ (pow t_m 3.0) (* l l))))))))

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 <= 4.3e-90) {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	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 <= 4.3d-90) then
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m))))) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    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 <= 4.3e-90) {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	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 <= 4.3e-90:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	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 <= 4.3e-90)
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	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 <= 4.3e-90)
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) * (k / t_m))))) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	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, 4.3e-90], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.3000000000000002e-90

   1. Initial program 49.1%

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

      1. associate-/r* 49.1%

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

      2. +-commutative 49.1%

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

      3. unpow2 49.1%

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

      4. sqr-neg 49.1%

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

      5. distribute-frac-neg2 49.1%

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

      6. distribute-frac-neg2 49.1%

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

      7. unpow2 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-*l* 44.3%

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

      10. associate-*l/ 44.4%

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

      11. associate-/r/ 44.2%

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

      12. +-commutative 44.2%

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

      13. associate-+r+ 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in t around 0 63.3%

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

      1. associate-*r/ 63.3%

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

      2. times-frac 62.5%

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

      3. associate-/r* 64.0%

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

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 50.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. pow2 50.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. add-sqr-sqrt 34.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right)} \]

      3. sqrt-div 14.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{\ell \cdot \ell}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      4. sqrt-prod 7.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      5. add-sqr-sqrt 8.5%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      6. *-commutative 8.5%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{\color{blue}{t \cdot {k}^{4}}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-prod 8.5%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      8. sqrt-pow1 8.5%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      9. metadata-eval 8.5%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{\color{blue}{2}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      10. sqrt-div 8.5%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \ell}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      11. sqrt-prod 7.7%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. add-sqr-sqrt 15.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. *-commutative 15.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{\color{blue}{t \cdot {k}^{4}}}}\right) \]

      14. sqrt-prod 15.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}}\right) \]

      15. sqrt-pow1 15.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}\right) \]

      16. metadata-eval 15.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot {k}^{\color{blue}{2}}}\right) \]

   10. Applied egg-rr 15.4%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot {k}^{2}}\right)} \]
   11. Step-by-step derivation

       1. unpow2 15.4%

          \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}}\right)}^{2}} \]

       2. *-commutative 15.4%

          \[\leadsto 2 \cdot {\left(\frac{\ell}{\color{blue}{{k}^{2} \cdot \sqrt{t}}}\right)}^{2} \]

   12. Simplified 15.4%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   if 4.3000000000000002e-90 < t

   1. Initial program 82.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 82.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

   6. Applied egg-rr 82.0%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6e-25)
    (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (/ (pow t_m 3.0) l) l)))))))

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 <= 6e-25) {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t_m, 3.0) / l) / l)));
	}
	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 <= 6d-25) then
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t_m ** 3.0d0) / l) / l)))
    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 <= 6e-25) {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l)));
	}
	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 <= 6e-25:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l)))
	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 <= 6e-25)
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	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 <= 6e-25)
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t_m ^ 3.0) / l) / l)));
	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, 6e-25], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.9999999999999995e-25

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 51.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

      3. unpow2 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

      4. sqr-neg 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

      5. distribute-frac-neg2 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

      6. distribute-frac-neg2 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

      7. unpow2 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

      8. +-commutative 51.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-*l* 46.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 46.8%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 46.6%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      12. +-commutative 46.6%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

      13. associate-+r+ 46.6%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 65.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 65.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 64.3%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. associate-/r* 65.7%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   7. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 51.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. pow2 51.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. add-sqr-sqrt 36.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right)} \]

      3. sqrt-div 17.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{\ell \cdot \ell}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      4. sqrt-prod 8.8%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      5. add-sqr-sqrt 10.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      6. *-commutative 10.6%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{\color{blue}{t \cdot {k}^{4}}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-prod 10.6%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      8. sqrt-pow1 10.6%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      9. metadata-eval 10.6%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{\color{blue}{2}}} \cdot \sqrt{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}}\right) \]

      10. sqrt-div 10.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \ell}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      11. sqrt-prod 9.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. add-sqr-sqrt 18.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. *-commutative 18.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{\color{blue}{t \cdot {k}^{4}}}}\right) \]

      14. sqrt-prod 18.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}}\right) \]

      15. sqrt-pow1 18.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}\right) \]

      16. metadata-eval 18.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot {k}^{\color{blue}{2}}}\right) \]

   10. Applied egg-rr 18.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}} \cdot \frac{\ell}{\sqrt{t} \cdot {k}^{2}}\right)} \]
   11. Step-by-step derivation

       1. unpow2 18.6%

          \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}}\right)}^{2}} \]

       2. *-commutative 18.6%

          \[\leadsto 2 \cdot {\left(\frac{\ell}{\color{blue}{{k}^{2} \cdot \sqrt{t}}}\right)}^{2} \]

   12. Simplified 18.6%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   if 5.9999999999999995e-25 < t

   1. Initial program 81.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 83.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. div-inv 83.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. add-cube-cbrt 83.4%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right)} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. associate-*l* 83.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. pow2 83.4%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. cbrt-div 83.3%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. rem-cbrt-cube 83.4%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. cbrt-div 83.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. rem-cbrt-cube 91.4%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 91.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{1}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 89.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow-plus 89.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. metadata-eval 89.8%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. add-cbrt-cube 83.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. unpow3 83.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\sqrt[3]{\color{blue}{{t}^{3}}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. cbrt-div 83.4%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. pow3 83.4%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right)} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. add-cube-cbrt 83.5%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. div-inv 83.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Applied egg-rr 83.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   9. Taylor expanded in k around 0 80.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 80.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   11. Simplified 80.4%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.8e-21)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (/ (pow t_m 3.0) l) l))))
    (* 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) {
	double tmp;
	if (k <= 3.8e-21) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	}
	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 (k <= 3.8d-21) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t_m ** 3.0d0) / l) / l)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    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 (k <= 3.8e-21) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	}
	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 k <= 3.8e-21:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	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 (k <= 3.8e-21)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	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 (k <= 3.8e-21)
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t_m ^ 3.0) / l) / l)));
	else
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	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[k, 3.8e-21], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.7999999999999998e-21

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 66.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. div-inv 66.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. add-cube-cbrt 66.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right)} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. associate-*l* 66.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. pow2 66.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. cbrt-div 66.2%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. rem-cbrt-cube 66.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. cbrt-div 66.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. rem-cbrt-cube 73.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{1}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 70.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow-plus 70.6%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. metadata-eval 70.6%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. add-cbrt-cube 66.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. unpow3 66.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\sqrt[3]{\color{blue}{{t}^{3}}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. cbrt-div 66.2%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}}^{3} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. pow3 66.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right)} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. add-cube-cbrt 66.3%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. div-inv 66.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Applied egg-rr 66.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   9. Taylor expanded in k around 0 64.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 64.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   11. Simplified 64.8%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 3.7999999999999998e-21 < k

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

      3. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

      4. sqr-neg 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

      5. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

      6. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

      7. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

      8. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-*l* 50.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 50.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 50.3%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      12. +-commutative 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

      13. associate-+r+ 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 72.8%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 70.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. associate-/r* 70.5%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 57.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. pow2 57.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-un-lft-identity 57.0%

         \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

      3. associate-/r* 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}}\right) \]

      4. pow2 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\color{blue}{{\ell}^{2}}}{{k}^{4}}}{t}\right) \]

   10. Applied egg-rr 57.1%

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 57.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

       2. associate-/l/ 57.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{4}}} \]

       3. associate-/r* 58.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   12. Simplified 58.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-20)
    (/ 2.0 (* (/ (/ (pow t_m 3.0) l) l) (* 2.0 (pow k 2.0))))
    (* 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) {
	double tmp;
	if (k <= 2.2e-20) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * (2.0 * pow(k, 2.0)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	}
	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 (k <= 2.2d-20) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) / l) * (2.0d0 * (k ** 2.0d0)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    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 (k <= 2.2e-20) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * (2.0 * Math.pow(k, 2.0)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	}
	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 k <= 2.2e-20:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) / l) * (2.0 * math.pow(k, 2.0)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	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 (k <= 2.2e-20)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(2.0 * (k ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	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 (k <= 2.2e-20)
		tmp = 2.0 / ((((t_m ^ 3.0) / l) / l) * (2.0 * (k ^ 2.0)));
	else
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	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[k, 2.2e-20], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.19999999999999991e-20

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 57.2%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   if 2.19999999999999991e-20 < k

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

      3. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

      4. sqr-neg 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

      5. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

      6. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

      7. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

      8. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-*l* 50.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 50.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 50.3%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      12. +-commutative 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

      13. associate-+r+ 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 72.8%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 70.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. associate-/r* 70.5%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 57.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. pow2 57.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-un-lft-identity 57.0%

         \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

      3. associate-/r* 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}}\right) \]

      4. pow2 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\color{blue}{{\ell}^{2}}}{{k}^{4}}}{t}\right) \]

   10. Applied egg-rr 57.1%

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 57.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

       2. associate-/l/ 57.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{4}}} \]

       3. associate-/r* 58.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   12. Simplified 58.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e-20)
    (/ 2.0 (/ (* (/ (pow t_m 3.0) l) (* 2.0 (pow k 2.0))) l))
    (* 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) {
	double tmp;
	if (k <= 1.45e-20) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) * (2.0 * pow(k, 2.0))) / l);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	}
	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 (k <= 1.45d-20) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) * (2.0d0 * (k ** 2.0d0))) / l)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    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 (k <= 1.45e-20) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) * (2.0 * Math.pow(k, 2.0))) / l);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	}
	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 k <= 1.45e-20:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) * (2.0 * math.pow(k, 2.0))) / l)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	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 (k <= 1.45e-20)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(2.0 * (k ^ 2.0))) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	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 (k <= 1.45e-20)
		tmp = 2.0 / ((((t_m ^ 3.0) / l) * (2.0 * (k ^ 2.0))) / l);
	else
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	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[k, 1.45e-20], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-20

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 57.2%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 57.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 57.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   if 1.45e-20 < k

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

      3. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

      4. sqr-neg 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

      5. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

      6. distribute-frac-neg2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

      7. unpow2 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

      8. +-commutative 50.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-*l* 50.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 50.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 50.3%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      12. +-commutative 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

      13. associate-+r+ 50.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 72.8%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 70.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. associate-/r* 70.5%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 57.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. pow2 57.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-un-lft-identity 57.0%

         \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

      3. associate-/r* 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}}\right) \]

      4. pow2 57.1%

         \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\color{blue}{{\ell}^{2}}}{{k}^{4}}}{t}\right) \]

   10. Applied egg-rr 57.1%

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 57.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

       2. associate-/l/ 57.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{4}}} \]

       3. associate-/r* 58.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   12. Simplified 58.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot {k}^{-4}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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(Float64((l ^ 2.0) / 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[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 58.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. +-commutative 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

   3. unpow2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

   4. sqr-neg 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

   5. distribute-frac-neg2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

   6. distribute-frac-neg2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

   7. unpow2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

   8. +-commutative 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

   9. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   10. associate-*l/ 53.4%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   11. associate-/r/ 53.3%

       \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   12. +-commutative 53.3%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

   13. associate-+r+ 53.3%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

3. Simplified 53.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 64.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 64.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. times-frac 64.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   3. associate-/r* 66.1%

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

7. Simplified 66.1%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

8. Taylor expanded in k around 0 53.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. pow2 53.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-un-lft-identity 53.5%

      \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

   3. associate-/r* 53.9%

      \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}}\right) \]

   4. pow2 53.9%

      \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\color{blue}{{\ell}^{2}}}{{k}^{4}}}{t}\right) \]

10. Applied egg-rr 53.9%

    \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 53.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

    2. associate-/l/ 53.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{4}}} \]

    3. associate-/r* 55.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

12. Simplified 55.0%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
13. Step-by-step derivation

    1. div-inv 55.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

    2. pow-flip 55.0%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \]

    3. metadata-eval 55.0%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right) \]

14. Applied egg-rr 55.0%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} \]

15. Final simplification 55.0%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right) \]
16. Add Preprocessing

Alternative 19: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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(Float64((l ^ 2.0) / 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[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 58.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. +-commutative 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

   3. unpow2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

   4. sqr-neg 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

   5. distribute-frac-neg2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

   6. distribute-frac-neg2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

   7. unpow2 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

   8. +-commutative 58.4%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

   9. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   10. associate-*l/ 53.4%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   11. associate-/r/ 53.3%

       \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   12. +-commutative 53.3%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

   13. associate-+r+ 53.3%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

3. Simplified 53.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 64.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 64.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. times-frac 64.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   3. associate-/r* 66.1%

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

7. Simplified 66.1%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

8. Taylor expanded in k around 0 53.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. pow2 53.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-un-lft-identity 53.5%

      \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

   3. associate-/r* 53.9%

      \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}}\right) \]

   4. pow2 53.9%

      \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\color{blue}{{\ell}^{2}}}{{k}^{4}}}{t}\right) \]

10. Applied egg-rr 53.9%

    \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 53.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

    2. associate-/l/ 53.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{4}}} \]

    3. associate-/r* 55.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

12. Simplified 55.0%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

13. Final simplification 55.0%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))