Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 90.0%

Time: 17.0s

Alternatives: 17

Speedup: 24.8×

• 27.4% 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: 54.3% 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: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot 2}}{t\_m \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= t_m 3.3e-11)
    (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
    (*
     (pow (/ (cbrt (* l 2.0)) (* t_m (* (cbrt (tan k)) (cbrt (sin k))))) 3.0)
     (/ l (+ 2.0 (pow (/ k t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-11) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else {
		tmp = pow((cbrt((l * 2.0)) / (t_m * (cbrt(tan(k)) * cbrt(sin(k))))), 3.0) * (l / (2.0 + pow((k / t_m), 2.0)));
	}
	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 <= 3.3e-11) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else {
		tmp = Math.pow((Math.cbrt((l * 2.0)) / (t_m * (Math.cbrt(Math.tan(k)) * Math.cbrt(Math.sin(k))))), 3.0) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-11)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	else
		tmp = Float64((Float64(cbrt(Float64(l * 2.0)) / Float64(t_m * Float64(cbrt(tan(k)) * cbrt(sin(k))))) ^ 3.0) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	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, 3.3e-11], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3000000000000002e-11

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 3.3000000000000002e-11 < t

   1. Initial program 74.0%

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

   3. Simplified 66.9%

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

      1. associate-*r* 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. times-frac 72.6%

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

      4. associate-*r* 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. /-rgt-identity 81.1%

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

      2. *-commutative 81.1%

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

      3. *-commutative 81.1%

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

   8. Simplified 81.1%

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

      1. add-cube-cbrt 81.0%

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

      2. pow3 81.1%

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

      3. associate-*r/ 81.1%

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

      4. cbrt-div 81.1%

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

      5. *-commutative 81.1%

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

      6. associate-*l* 72.7%

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

      7. cbrt-prod 72.6%

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

      8. unpow3 72.6%

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

      9. add-cbrt-cube 81.7%

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

   10. Applied egg-rr 81.7%

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

       1. *-commutative 81.7%

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

       2. cbrt-prod 95.0%

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

   12. Applied egg-rr 95.0%

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

Alternative 2: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t\_2 + 1\right)\right)} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\ell}{2 + t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (<=
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))
           (+ 1.0 (+ t_2 1.0))))
         2e+270)
      (* (/ l (+ 2.0 t_2)) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
      (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+270) {
		tmp = (l / (2.0 + t_2)) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	}
	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 = (k / t_m) ** 2.0d0
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))) * (1.0d0 + (t_2 + 1.0d0)))) <= 2d+270) then
        tmp = (l / (2.0d0 + t_2)) * (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0)))))
    else
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) ** 2.0d0) * (cos(k) / t_m)
    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((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+270) {
		tmp = (l / (2.0 + t_2)) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	}
	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((k / t_m), 2.0)
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+270:
		tmp = (l / (2.0 + t_2)) * (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0)))))
	else:
		tmp = math.pow((l * (math.sqrt(2.0) / (k * math.sin(k)))), 2.0) * (math.cos(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 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0)))) <= 2e+270)
		tmp = Float64(Float64(l / Float64(2.0 + t_2)) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	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 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+270)
		tmp = (l / (2.0 + t_2)) * (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0)))));
	else
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) ^ 2.0) * (cos(k) / t_m);
	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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+270], N[(N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e270

   1. Initial program 82.2%

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

   3. Simplified 74.9%

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

      1. associate-*r* 75.7%

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

      2. *-un-lft-identity 75.7%

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

      3. times-frac 77.2%

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

      4. associate-*r* 85.1%

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

   6. Applied egg-rr 85.1%

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

      1. /-rgt-identity 85.1%

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

      2. *-commutative 85.1%

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

      3. *-commutative 85.1%

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

   8. Simplified 85.1%

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

   if 2.0000000000000001e270 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 24.6%

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

   3. Simplified 24.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. pow2 24.6%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in t around 0 60.5%

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

      1. unpow-prod-down 58.3%

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

      2. times-frac 58.3%

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

      3. pow2 58.3%

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

      4. add-sqr-sqrt 80.7%

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

   9. Applied egg-rr 80.7%

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

   10. Taylor expanded in l around 0 80.7%

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

       1. associate-/l* 80.7%

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

   12. Simplified 80.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot t\_2\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-10)
      (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
      (if (<= t_m 5.6e+102)
        (* (/ l t_2) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (/
         2.0
         (*
          (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)
          (* (sin k) (* (tan 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 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.6e-10) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (sin(k) * (tan(k) * 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 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.6e-10) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = (l / t_2) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (Math.sin(k) * (Math.tan(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(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.6e-10)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(sin(k) * Float64(tan(k) * 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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-10], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5999999999999999e-10

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 1.5999999999999999e-10 < t < 5.60000000000000037e102

   1. Initial program 88.4%

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

   3. Simplified 84.4%

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

      1. associate-*r* 88.2%

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

      2. *-un-lft-identity 88.2%

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

      3. times-frac 88.2%

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

      4. associate-*r* 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. /-rgt-identity 95.8%

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

      2. *-commutative 95.8%

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

      3. *-commutative 95.8%

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

   8. Simplified 95.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 66.3%

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

   3. Simplified 64.4%

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

      1. unpow2 64.4%

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

      2. clear-num 64.4%

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

      3. un-div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

      1. add-cube-cbrt 64.4%

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

      2. pow3 64.4%

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

   8. Applied egg-rr 84.8%

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

      1. cube-prod 84.6%

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

      2. rem-cube-cbrt 84.6%

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

      3. associate-*l* 84.6%

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

   10. Simplified 84.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\sqrt[3]{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({t\_2}^{2} \cdot \frac{t\_2}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (cbrt l))))
   (*
    t_s
    (if (<= t_m 5.6e-11)
      (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
      (if (<= t_m 5.6e+102)
        (*
         (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
         (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (/
         2.0
         (*
          (* (pow t_2 2.0) (/ t_2 l))
          (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m 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 t_2 = t_m / cbrt(l);
	double tmp;
	if (t_m <= 5.6e-11) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / ((pow(t_2, 2.0) * (t_2 / l)) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / 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 t_2 = t_m / Math.cbrt(l);
	double tmp;
	if (t_m <= 5.6e-11) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / ((Math.pow(t_2, 2.0) * (t_2 / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / cbrt(l))
	tmp = 0.0
	if (t_m <= 5.6e-11)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64((t_2 ^ 2.0) * Float64(t_2 / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / 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_] := Block[{t$95$2 = N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.6e-11], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * 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]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.6e-11

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 5.6e-11 < t < 5.60000000000000037e102

   1. Initial program 88.4%

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

   3. Simplified 84.4%

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

      1. associate-*r* 88.2%

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

      2. *-un-lft-identity 88.2%

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

      3. times-frac 88.2%

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

      4. associate-*r* 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. /-rgt-identity 95.8%

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

      2. *-commutative 95.8%

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

      3. *-commutative 95.8%

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

   8. Simplified 95.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 66.3%

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

   3. Simplified 64.4%

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

      1. unpow2 64.4%

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

      2. clear-num 64.4%

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

      3. un-div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

      1. add-cube-cbrt 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 64.4%

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

      4. pow2 64.4%

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

      5. cbrt-div 64.4%

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

      6. rem-cbrt-cube 64.4%

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

      7. cbrt-div 64.4%

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

      8. rem-cbrt-cube 80.8%

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

   8. Applied egg-rr 80.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right)\\ \end{array} \end{array} \]

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 5.6e-11)
    (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
    (*
     (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
     (* l (/ 2.0 (* (tan k) (pow (* t_m (cbrt (sin k))) 3.0))))))))

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 <= 5.6e-11) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * pow((t_m * cbrt(sin(k))), 3.0))));
	}
	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 <= 5.6e-11) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))));
	}
	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 <= 5.6e-11)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k) * (Float64(t_m * cbrt(sin(k))) ^ 3.0)))));
	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, 5.6e-11], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.6e-11

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 5.6e-11 < t

   1. Initial program 74.0%

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

   3. Simplified 66.9%

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

      1. associate-*r* 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. times-frac 72.6%

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

      4. associate-*r* 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. /-rgt-identity 81.1%

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

      2. *-commutative 81.1%

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

      3. *-commutative 81.1%

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

   8. Simplified 81.1%

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

      1. *-commutative 81.1%

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

      2. add-cube-cbrt 81.0%

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

      3. pow3 81.0%

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

      4. cbrt-prod 80.9%

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

      5. unpow3 81.0%

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

      6. add-cbrt-cube 86.4%

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

   10. Applied egg-rr 86.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102} \lor \neg \left(t\_m \leq 3.2 \cdot 10^{+205}\right):\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= t_m 4e-11)
    (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
    (if (or (<= t_m 5.6e+102) (not (<= t_m 3.2e+205)))
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m 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 <= 4e-11) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else if ((t_m <= 5.6e+102) || !(t_m <= 3.2e+205)) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / 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 <= 4d-11) then
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) ** 2.0d0) * (cos(k) / t_m)
    else if ((t_m <= 5.6d+102) .or. (.not. (t_m <= 3.2d+205))) then
        tmp = (l / (2.0d0 + ((k / t_m) ** 2.0d0))) * (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0)))))
    else
        tmp = 2.0d0 / (((sin(k) * tan(k)) * (2.0d0 + ((k / t_m) / (t_m / 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 <= 4e-11) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else if ((t_m <= 5.6e+102) || !(t_m <= 3.2e+205)) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / 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 <= 4e-11:
		tmp = math.pow((l * (math.sqrt(2.0) / (k * math.sin(k)))), 2.0) * (math.cos(k) / t_m)
	elif (t_m <= 5.6e+102) or not (t_m <= 3.2e+205):
		tmp = (l / (2.0 + math.pow((k / t_m), 2.0))) * (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0)))))
	else:
		tmp = 2.0 / (((math.sin(k) * math.tan(k)) * (2.0 + ((k / t_m) / (t_m / 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 <= 4e-11)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	elseif ((t_m <= 5.6e+102) || !(t_m <= 3.2e+205))
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / 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 <= 4e-11)
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) ^ 2.0) * (cos(k) / t_m);
	elseif ((t_m <= 5.6e+102) || ~((t_m <= 3.2e+205)))
		tmp = (l / (2.0 + ((k / t_m) ^ 2.0))) * (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0)))));
	else
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / 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, 4e-11], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 5.6e+102], N[Not[LessEqual[t$95$m, 3.2e+205]], $MachinePrecision]], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(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] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.99999999999999976e-11

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 3.99999999999999976e-11 < t < 5.60000000000000037e102 or 3.19999999999999996e205 < t

   1. Initial program 75.9%

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

   3. Simplified 66.2%

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

      1. associate-*r* 74.0%

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

      2. *-un-lft-identity 74.0%

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

      3. times-frac 74.0%

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

      4. associate-*r* 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. /-rgt-identity 85.3%

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

      2. *-commutative 85.3%

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

      3. *-commutative 85.3%

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

   8. Simplified 85.3%

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

   if 5.60000000000000037e102 < t < 3.19999999999999996e205

   1. Initial program 68.7%

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

   3. Simplified 68.9%

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

      1. unpow2 68.9%

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

      2. clear-num 68.9%

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

      3. un-div-inv 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. add-sqr-sqrt 68.9%

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

      2. pow2 68.9%

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

      3. associate-/r* 68.7%

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

      4. sqrt-div 68.7%

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

      5. sqrt-pow1 94.9%

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

      6. metadata-eval 94.9%

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

      7. sqrt-prod 68.1%

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

      8. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102} \lor \neg \left(t \leq 3.2 \cdot 10^{+205}\right):\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t\_m}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.3e-10)
    (* (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0) (/ (cos k) t_m))
    (if (<= t_m 7.2e+164)
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))
        (* (/ (pow t_m 2.0) l) (/ t_m l))))
      (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k 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 <= 1.3e-10) {
		tmp = pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0) * (cos(k) / t_m);
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.3e-10) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0) * (Math.cos(k) / t_m);
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.3e-10)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0) * Float64(cos(k) / t_m));
	elseif (t_m <= 7.2e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * 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, 1.3e-10], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+164], N[(2.0 / N[(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] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.29999999999999991e-10

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. unpow-prod-down 38.5%

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

      2. times-frac 38.5%

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

      3. pow2 38.5%

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

      4. add-sqr-sqrt 80.0%

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

   9. Applied egg-rr 80.0%

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

   10. Taylor expanded in l around 0 80.0%

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

       1. associate-/l* 80.1%

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

   12. Simplified 80.1%

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

   if 1.29999999999999991e-10 < t < 7.19999999999999981e164

   1. Initial program 76.9%

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

   3. Simplified 76.7%

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

      1. unpow2 76.7%

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

      2. clear-num 76.7%

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

      3. un-div-inv 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. associate-/r* 73.9%

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

      2. unpow3 73.9%

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

      3. times-frac 88.4%

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

      4. pow2 88.4%

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

   8. Applied egg-rr 88.4%

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

   if 7.19999999999999981e164 < t

   1. Initial program 71.4%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in k around 0 68.9%

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

      1. unpow2 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-/r* 60.5%

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

      2. unpow3 60.5%

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

      3. times-frac 68.9%

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

      4. pow2 68.9%

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

   9. Applied egg-rr 68.9%

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

       1. add-cube-cbrt 68.9%

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

       2. pow3 68.9%

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

       3. frac-times 60.5%

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

       4. unpow2 60.5%

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

       5. unpow3 60.5%

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

       6. cbrt-div 60.5%

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

       7. unpow3 60.5%

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

       8. add-cbrt-cube 63.4%

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

       9. cbrt-prod 76.4%

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

       10. pow2 76.4%

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

   11. Applied egg-rr 76.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.32 \cdot 10^{-10}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2} \cdot \frac{\ell}{k}}{\sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.32e-10)
    (* (/ (cos k) t_m) (pow (/ (* (sqrt 2.0) (/ l k)) (sin k)) 2.0))
    (if (<= t_m 7.2e+164)
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))
        (* (/ (pow t_m 2.0) l) (/ t_m l))))
      (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k 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 <= 1.32e-10) {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) * (l / k)) / sin(k)), 2.0);
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.32e-10) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) * (l / k)) / Math.sin(k)), 2.0);
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.32e-10)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) * Float64(l / k)) / sin(k)) ^ 2.0));
	elseif (t_m <= 7.2e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * 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, 1.32e-10], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+164], N[(2.0 / N[(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] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.3200000000000001e-10

   1. Initial program 45.6%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   6. Applied egg-rr 33.3%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. *-un-lft-identity 40.0%

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

      2. *-commutative 40.0%

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

      3. unpow-prod-down 38.5%

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

      4. pow2 38.5%

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

      5. add-sqr-sqrt 80.0%

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

      6. times-frac 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-lft-identity 80.0%

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

       2. associate-*r/ 80.0%

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

   11. Simplified 80.0%

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

   if 1.3200000000000001e-10 < t < 7.19999999999999981e164

   1. Initial program 76.9%

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

   3. Simplified 76.7%

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

      1. unpow2 76.7%

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

      2. clear-num 76.7%

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

      3. un-div-inv 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. associate-/r* 73.9%

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

      2. unpow3 73.9%

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

      3. times-frac 88.4%

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

      4. pow2 88.4%

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

   8. Applied egg-rr 88.4%

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

   if 7.19999999999999981e164 < t

   1. Initial program 71.4%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in k around 0 68.9%

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

      1. unpow2 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-/r* 60.5%

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

      2. unpow3 60.5%

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

      3. times-frac 68.9%

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

      4. pow2 68.9%

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

   9. Applied egg-rr 68.9%

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

       1. add-cube-cbrt 68.9%

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

       2. pow3 68.9%

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

       3. frac-times 60.5%

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

       4. unpow2 60.5%

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

       5. unpow3 60.5%

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

       6. cbrt-div 60.5%

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

       7. unpow3 60.5%

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

       8. add-cbrt-cube 63.4%

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

       9. cbrt-prod 76.4%

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

       10. pow2 76.4%

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

   11. Applied egg-rr 76.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.32 \cdot 10^{-10}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2} \cdot \frac{\ell}{k}}{\sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.68 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.9e-31)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (pow k 2.0))) 2.0))
    (if (<= t_m 1.68e+81)
      (/ 2.0 (* (/ (* k (pow t_m 3.0)) (pow l 2.0)) (* 2.0 k)))
      (if (<= t_m 7.2e+164)
        (/
         2.0
         (*
          (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))
          (* (/ (pow t_m 2.0) l) (/ t_m l))))
        (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k 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 <= 1.9e-31) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / pow(k, 2.0))), 2.0);
	} else if (t_m <= 1.68e+81) {
		tmp = 2.0 / (((k * pow(t_m, 3.0)) / pow(l, 2.0)) * (2.0 * k));
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.9e-31) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))), 2.0);
	} else if (t_m <= 1.68e+81) {
		tmp = 2.0 / (((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)) * (2.0 * k));
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 1.9e-31)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) ^ 2.0));
	elseif (t_m <= 1.68e+81)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)) * Float64(2.0 * k)));
	elseif (t_m <= 7.2e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * 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, 1.9e-31], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.68e+81], N[(2.0 / N[(N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+164], N[(2.0 / N[(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] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.9e-31

   1. Initial program 45.5%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 27.4%

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

      2. pow2 27.4%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in t around 0 38.7%

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

      1. unpow-prod-down 37.2%

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

      2. times-frac 37.1%

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

      3. pow2 37.1%

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

      4. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in k around 0 62.2%

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

       1. associate-/l* 62.3%

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

   12. Simplified 62.3%

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

   if 1.9e-31 < t < 1.68000000000000001e81

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in k around 0 77.6%

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

   6. Taylor expanded in k around 0 82.4%

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

   if 1.68000000000000001e81 < t < 7.19999999999999981e164

   1. Initial program 69.1%

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

   3. Simplified 63.4%

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

      1. unpow2 63.4%

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

      2. clear-num 63.4%

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

      3. un-div-inv 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. associate-/r* 63.2%

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

      2. unpow3 63.1%

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

      3. times-frac 88.1%

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

      4. pow2 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 7.19999999999999981e164 < t

   1. Initial program 71.4%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in k around 0 68.9%

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

      1. unpow2 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-/r* 60.5%

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

      2. unpow3 60.5%

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

      3. times-frac 68.9%

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

      4. pow2 68.9%

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

   9. Applied egg-rr 68.9%

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

       1. add-cube-cbrt 68.9%

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

       2. pow3 68.9%

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

       3. frac-times 60.5%

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

       4. unpow2 60.5%

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

       5. unpow3 60.5%

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

       6. cbrt-div 60.5%

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

       7. unpow3 60.5%

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

       8. add-cbrt-cube 63.4%

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

       9. cbrt-prod 76.4%

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

       10. pow2 76.4%

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

   11. Applied egg-rr 76.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.68 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2.6e-27)
    (* (/ (cos k) t_m) (pow (* (/ l k) (/ (sqrt 2.0) k)) 2.0))
    (if (<= t_m 1.95e+80)
      (/ 2.0 (* (/ (* k (pow t_m 3.0)) (pow l 2.0)) (* 2.0 k)))
      (if (<= t_m 7.2e+164)
        (/
         2.0
         (*
          (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))
          (* (/ (pow t_m 2.0) l) (/ t_m l))))
        (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k 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 <= 2.6e-27) {
		tmp = (cos(k) / t_m) * pow(((l / k) * (sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 1.95e+80) {
		tmp = 2.0 / (((k * pow(t_m, 3.0)) / pow(l, 2.0)) * (2.0 * k));
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 2.6e-27) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((l / k) * (Math.sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 1.95e+80) {
		tmp = 2.0 / (((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)) * (2.0 * k));
	} else if (t_m <= 7.2e+164) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 2.6e-27)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(l / k) * Float64(sqrt(2.0) / k)) ^ 2.0));
	elseif (t_m <= 1.95e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)) * Float64(2.0 * k)));
	elseif (t_m <= 7.2e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * 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, 2.6e-27], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+80], N[(2.0 / N[(N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+164], N[(2.0 / N[(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] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.60000000000000017e-27

   1. Initial program 45.5%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 27.4%

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

      2. pow2 27.4%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in t around 0 38.7%

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

      1. unpow-prod-down 37.2%

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

      2. times-frac 37.1%

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

      3. pow2 37.1%

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

      4. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in k around 0 62.2%

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

   if 2.60000000000000017e-27 < t < 1.94999999999999999e80

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in k around 0 77.6%

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

   6. Taylor expanded in k around 0 82.4%

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

   if 1.94999999999999999e80 < t < 7.19999999999999981e164

   1. Initial program 69.1%

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

   3. Simplified 63.4%

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

      1. unpow2 63.4%

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

      2. clear-num 63.4%

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

      3. un-div-inv 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. associate-/r* 63.2%

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

      2. unpow3 63.1%

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

      3. times-frac 88.1%

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

      4. pow2 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 7.19999999999999981e164 < t

   1. Initial program 71.4%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in k around 0 68.9%

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

      1. unpow2 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-/r* 60.5%

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

      2. unpow3 60.5%

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

      3. times-frac 68.9%

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

      4. pow2 68.9%

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

   9. Applied egg-rr 68.9%

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

       1. add-cube-cbrt 68.9%

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

       2. pow3 68.9%

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

       3. frac-times 60.5%

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

       4. unpow2 60.5%

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

       5. unpow3 60.5%

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

       6. cbrt-div 60.5%

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

       7. unpow3 60.5%

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

       8. add-cbrt-cube 63.4%

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

       9. cbrt-prod 76.4%

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

       10. pow2 76.4%

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

   11. Applied egg-rr 76.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2.7e-32)
    (* (/ (cos k) t_m) (pow (* (/ l k) (/ (sqrt 2.0) k)) 2.0))
    (if (<= t_m 2.4e+102)
      (/ 2.0 (* (/ (* k (pow t_m 3.0)) (pow l 2.0)) (* 2.0 k)))
      (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k 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 <= 2.7e-32) {
		tmp = (cos(k) / t_m) * pow(((l / k) * (sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 2.4e+102) {
		tmp = 2.0 / (((k * pow(t_m, 3.0)) / pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 2.7e-32) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((l / k) * (Math.sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 2.4e+102) {
		tmp = 2.0 / (((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * 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 <= 2.7e-32)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(l / k) * Float64(sqrt(2.0) / k)) ^ 2.0));
	elseif (t_m <= 2.4e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * 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, 2.7e-32], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+102], N[(2.0 / N[(N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.69999999999999981e-32

   1. Initial program 45.5%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 27.4%

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

      2. pow2 27.4%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in t around 0 38.7%

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

      1. unpow-prod-down 37.2%

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

      2. times-frac 37.1%

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

      3. pow2 37.1%

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

      4. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in k around 0 62.2%

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

   if 2.69999999999999981e-32 < t < 2.39999999999999994e102

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in k around 0 79.9%

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

   6. Taylor expanded in k around 0 83.5%

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

   if 2.39999999999999994e102 < t

   1. Initial program 66.3%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in k around 0 64.4%

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

      1. unpow2 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. associate-/r* 57.6%

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

      2. unpow3 57.6%

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

      3. times-frac 72.8%

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

      4. pow2 72.8%

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

   9. Applied egg-rr 70.7%

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

       1. add-cube-cbrt 70.7%

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

       2. pow3 70.7%

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

       3. frac-times 57.6%

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

       4. unpow2 57.6%

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

       5. unpow3 57.6%

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

       6. cbrt-div 57.6%

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

       7. unpow3 57.6%

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

       8. add-cbrt-cube 66.3%

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

       9. cbrt-prod 76.8%

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

       10. pow2 76.8%

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

   11. Applied egg-rr 76.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 3.55 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2.35e-32)
    (* (/ (cos k) t_m) (pow (* (/ l k) (/ (sqrt 2.0) k)) 2.0))
    (if (<= t_m 3.55e+102)
      (/ 2.0 (* (/ (* k (pow t_m 3.0)) (pow l 2.0)) (* 2.0 k)))
      (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k 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 <= 2.35e-32) {
		tmp = (cos(k) / t_m) * pow(((l / k) * (sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 3.55e+102) {
		tmp = 2.0 / (((k * pow(t_m, 3.0)) / pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	}
	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 <= 2.35d-32) then
        tmp = (cos(k) / t_m) * (((l / k) * (sqrt(2.0d0) / k)) ** 2.0d0)
    else if (t_m <= 3.55d+102) then
        tmp = 2.0d0 / (((k * (t_m ** 3.0d0)) / (l ** 2.0d0)) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k * k)))
    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 <= 2.35e-32) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((l / k) * (Math.sqrt(2.0) / k)), 2.0);
	} else if (t_m <= 3.55e+102) {
		tmp = 2.0 / (((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	}
	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 <= 2.35e-32:
		tmp = (math.cos(k) / t_m) * math.pow(((l / k) * (math.sqrt(2.0) / k)), 2.0)
	elif t_m <= 3.55e+102:
		tmp = 2.0 / (((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)) * (2.0 * k))
	else:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * 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 <= 2.35e-32)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(l / k) * Float64(sqrt(2.0) / k)) ^ 2.0));
	elseif (t_m <= 3.55e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	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 <= 2.35e-32)
		tmp = (cos(k) / t_m) * (((l / k) * (sqrt(2.0) / k)) ^ 2.0);
	elseif (t_m <= 3.55e+102)
		tmp = 2.0 / (((k * (t_m ^ 3.0)) / (l ^ 2.0)) * (2.0 * k));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k * k)));
	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, 2.35e-32], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.55e+102], N[(2.0 / N[(N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-32

   1. Initial program 45.5%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 27.4%

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

      2. pow2 27.4%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in t around 0 38.7%

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

      1. unpow-prod-down 37.2%

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

      2. times-frac 37.1%

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

      3. pow2 37.1%

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

      4. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in k around 0 62.2%

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

   if 2.3500000000000001e-32 < t < 3.5499999999999999e102

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in k around 0 79.9%

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

   6. Taylor expanded in k around 0 83.5%

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

   if 3.5499999999999999e102 < t

   1. Initial program 66.3%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in k around 0 64.4%

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

      1. unpow2 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. add-sqr-sqrt 64.4%

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

      2. pow2 64.4%

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

      3. associate-/r* 57.6%

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

      4. sqrt-div 57.6%

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

      5. sqrt-pow1 68.2%

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

      6. metadata-eval 68.2%

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

      7. sqrt-prod 46.7%

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

      8. add-sqr-sqrt 76.9%

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

   9. Applied egg-rr 72.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.55 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 3.25 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 2e-36)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (if (<= t_m 3.25e+102)
      (/ 2.0 (* (/ (* k (pow t_m 3.0)) (pow l 2.0)) (* 2.0 k)))
      (/ 2.0 (* (* 2.0 (* k k)) (/ (* (pow t_m 2.0) (/ t_m 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 <= 2e-36) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else if (t_m <= 3.25e+102) {
		tmp = 2.0 / (((k * pow(t_m, 3.0)) / pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (t_m / 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 <= 2d-36) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else if (t_m <= 3.25d+102) then
        tmp = 2.0d0 / (((k * (t_m ** 3.0d0)) / (l ** 2.0d0)) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (t_m / 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 <= 2e-36) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else if (t_m <= 3.25e+102) {
		tmp = 2.0 / (((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (t_m / 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 <= 2e-36:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	elif t_m <= 3.25e+102:
		tmp = 2.0 / (((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)) * (2.0 * k))
	else:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (t_m / 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 <= 2e-36)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	elseif (t_m <= 3.25e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m / 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 <= 2e-36)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	elseif (t_m <= 3.25e+102)
		tmp = 2.0 / (((k * (t_m ^ 3.0)) / (l ^ 2.0)) * (2.0 * k));
	else
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (t_m / 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, 2e-36], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.25e+102], N[(2.0 / N[(N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.9999999999999999e-36

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in t around 0 66.0%

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

      1. associate-*r* 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in k around 0 52.8%

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

      1. associate-/l* 54.2%

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

   10. Simplified 54.2%

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

   if 1.9999999999999999e-36 < t < 3.2500000000000002e102

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in k around 0 77.9%

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

   6. Taylor expanded in k around 0 81.4%

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

   if 3.2500000000000002e102 < t

   1. Initial program 66.3%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in k around 0 64.4%

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

      1. unpow2 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. associate-/r* 57.6%

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

      2. unpow3 57.6%

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

      3. times-frac 72.8%

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

      4. pow2 72.8%

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

   9. Applied egg-rr 70.7%

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

       1. associate-*l/ 70.7%

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

   11. Applied egg-rr 70.7%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t\_m}\\ \end{array} \end{array} \]

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 8e+159)
    (/ 2.0 (* (* 2.0 (* k k)) (/ (* (pow t_m 2.0) (/ t_m l)) l)))
    (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 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 (k <= 8e+159) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	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 <= 8d+159) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (t_m / l)) / l))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    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 <= 8e+159) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	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 <= 8e+159:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (t_m / l)) / l))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / 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 (k <= 8e+159)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	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 <= 8e+159)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (t_m / l)) / l));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	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, 8e+159], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.9999999999999994e159

   1. Initial program 53.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. unpow2 57.6%

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

   7. Applied egg-rr 57.6%

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

      1. associate-/r* 48.4%

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

      2. unpow3 48.4%

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

      3. times-frac 59.0%

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

      4. pow2 59.0%

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

   9. Applied egg-rr 60.7%

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

       1. associate-*l/ 60.8%

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

   11. Applied egg-rr 60.8%

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

   if 7.9999999999999994e159 < k

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-sqr-sqrt 57.1%

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

      2. pow2 57.1%

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

   6. Applied egg-rr 49.1%

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

   7. Taylor expanded in t around 0 48.3%

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

      1. unpow-prod-down 45.8%

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

      2. times-frac 45.8%

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

      3. pow2 45.8%

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

      4. add-sqr-sqrt 91.2%

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

   9. Applied egg-rr 91.2%

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

   10. Taylor expanded in k around 0 77.0%

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

       1. times-frac 77.0%

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

       2. unpow2 77.0%

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

       3. rem-square-sqrt 77.0%

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

   12. Simplified 77.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \end{array} \]

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 (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k k))))))

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((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k * k))))
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((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))));
}

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((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k * k)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Simplified 53.8%

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

5. Taylor expanded in k around 0 57.5%

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

   1. unpow2 57.5%

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

7. Applied egg-rr 57.5%

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

   1. add-sqr-sqrt 29.5%

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

   2. pow2 29.5%

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

   3. associate-/r* 27.1%

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

   4. sqrt-div 27.1%

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

   5. sqrt-pow1 30.6%

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

   6. metadata-eval 30.6%

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

   7. sqrt-prod 19.1%

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

   8. add-sqr-sqrt 33.8%

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

9. Applied egg-rr 33.9%

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

Alternative 16: 57.4% accurate, 3.6× speedup

Math

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

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 (* (* 2.0 (* k k)) (/ (* (pow t_m 2.0) (/ t_m 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (t_m / l)) / l)));
}

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 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (t_m / l)) / l)))
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 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (t_m / l)) / l)));
}

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 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (t_m / l)) / l)))

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(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l))))
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 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (t_m / l)) / l)));
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[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Simplified 53.8%

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

5. Taylor expanded in k around 0 57.5%

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

   1. unpow2 57.5%

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

7. Applied egg-rr 57.5%

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

   1. associate-/r* 49.6%

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

   2. unpow3 49.6%

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

   3. times-frac 59.9%

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

   4. pow2 59.9%

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

9. Applied egg-rr 61.1%

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

    1. associate-*l/ 61.1%

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

11. Applied egg-rr 61.1%

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

12. Final simplification 61.1%

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

Alternative 17: 57.7% accurate, 24.8× speedup

Math

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

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 (* (* 2.0 (* k k)) (* (/ t_m l) (/ (* t_m 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
}

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 / ((2.0d0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))))
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 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
}

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 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))))

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(2.0 * Float64(k * k)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l)))))
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 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
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[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Simplified 53.8%

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

5. Taylor expanded in k around 0 57.5%

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

   1. unpow2 57.5%

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

7. Applied egg-rr 57.5%

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

   1. associate-/r* 49.6%

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

   2. unpow3 49.6%

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

   3. times-frac 59.9%

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

   4. pow2 59.9%

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

9. Applied egg-rr 61.1%

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

    1. unpow2 61.1%

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

11. Applied egg-rr 61.1%

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

12. Final simplification 61.1%

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

Reproduce

herbie shell --seed 2024136 
(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))))