Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 91.1%

Time: 20.4s

Alternatives: 26

Speedup: 3.7×

• 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: 54.5% 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: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.85 \cdot 10^{-246}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \]

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 3.85e-246)
    (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ (cos k) t_m))) 2.0)
    (pow
     (*
      (/ (/ (cbrt (/ 2.0 (tan k))) (cbrt (sin k))) t_m)
      (pow (cbrt (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))) 2.0))
     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 <= 3.85e-246) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((cos(k) / t_m))), 2.0);
	} else {
		tmp = pow((((cbrt((2.0 / tan(k))) / cbrt(sin(k))) / t_m) * pow(cbrt((l / hypot(1.0, hypot(1.0, (k / t_m))))), 2.0)), 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 <= 3.85e-246) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else {
		tmp = Math.pow((((Math.cbrt((2.0 / Math.tan(k))) / Math.cbrt(Math.sin(k))) / t_m) * Math.pow(Math.cbrt((l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m))))), 2.0)), 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 <= 3.85e-246)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(cbrt(Float64(2.0 / tan(k))) / cbrt(sin(k))) / t_m) * (cbrt(Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))) ^ 2.0)) ^ 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, 3.85e-246], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.8499999999999998e-246

   1. Initial program 64.9%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-div 34.5%

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

      3. sqrt-div 34.5%

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

   6. Applied egg-rr 29.7%

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

      1. unpow2 29.7%

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

   8. Simplified 33.9%

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

   9. Taylor expanded in k around inf 29.3%

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

       1. associate-/l* 29.3%

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

   11. Simplified 29.3%

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

   if 3.8499999999999998e-246 < t

   1. Initial program 51.4%

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

   3. Simplified 50.5%

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

      1. associate-*r* 54.7%

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

      2. add-sqr-sqrt 54.6%

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

      3. times-frac 54.7%

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

      4. div-inv 54.7%

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

      5. frac-times 54.7%

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

      6. metadata-eval 54.7%

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

   6. Applied egg-rr 54.7%

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

   8. Simplified 63.2%

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

      1. add-cube-cbrt 63.0%

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

      2. pow3 63.0%

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

   10. Applied egg-rr 81.9%

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

       1. cbrt-div 96.6%

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

   12. Applied egg-rr 96.6%

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

Alternative 2: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t_3 := \frac{2}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-235}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{-115}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{t\_3}{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\frac{t\_m \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \frac{\ell \cdot t\_3}{t\_2 \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{t\_3}}{\sqrt[3]{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\ell \cdot \sqrt{0.5}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m)))) (t_3 (/ 2.0 (tan k))))
   (*
    t_s
    (if (<= t_m 1.7e-235)
      (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 6.2e-115)
        (pow
         (* (/ (cbrt (/ t_3 (sin k))) t_m) (pow (cbrt (/ (* t_m l) k)) 2.0))
         3.0)
        (if (<= t_m 2.7e+85)
          (* (/ l t_2) (/ (* l t_3) (* t_2 (* (sin k) (pow t_m 3.0)))))
          (pow
           (*
            (/ (/ (cbrt t_3) (cbrt (sin k))) t_m)
            (pow (cbrt (* l (sqrt 0.5))) 2.0))
           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 t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double t_3 = 2.0 / tan(k);
	double tmp;
	if (t_m <= 1.7e-235) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 6.2e-115) {
		tmp = pow(((cbrt((t_3 / sin(k))) / t_m) * pow(cbrt(((t_m * l) / k)), 2.0)), 3.0);
	} else if (t_m <= 2.7e+85) {
		tmp = (l / t_2) * ((l * t_3) / (t_2 * (sin(k) * pow(t_m, 3.0))));
	} else {
		tmp = pow((((cbrt(t_3) / cbrt(sin(k))) / t_m) * pow(cbrt((l * sqrt(0.5))), 2.0)), 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 t_2 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_3 = 2.0 / Math.tan(k);
	double tmp;
	if (t_m <= 1.7e-235) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 6.2e-115) {
		tmp = Math.pow(((Math.cbrt((t_3 / Math.sin(k))) / t_m) * Math.pow(Math.cbrt(((t_m * l) / k)), 2.0)), 3.0);
	} else if (t_m <= 2.7e+85) {
		tmp = (l / t_2) * ((l * t_3) / (t_2 * (Math.sin(k) * Math.pow(t_m, 3.0))));
	} else {
		tmp = Math.pow((((Math.cbrt(t_3) / Math.cbrt(Math.sin(k))) / t_m) * Math.pow(Math.cbrt((l * Math.sqrt(0.5))), 2.0)), 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)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	t_3 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t_m <= 1.7e-235)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 6.2e-115)
		tmp = Float64(Float64(cbrt(Float64(t_3 / sin(k))) / t_m) * (cbrt(Float64(Float64(t_m * l) / k)) ^ 2.0)) ^ 3.0;
	elseif (t_m <= 2.7e+85)
		tmp = Float64(Float64(l / t_2) * Float64(Float64(l * t_3) / Float64(t_2 * Float64(sin(k) * (t_m ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(cbrt(t_3) / cbrt(sin(k))) / t_m) * (cbrt(Float64(l * sqrt(0.5))) ^ 2.0)) ^ 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_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.7e-235], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 6.2e-115], N[Power[N[(N[(N[Power[N[(t$95$3 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$m, 2.7e+85], N[(N[(l / t$95$2), $MachinePrecision] * N[(N[(l * t$95$3), $MachinePrecision] / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(N[Power[t$95$3, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.69999999999999986e-235

   1. Initial program 64.9%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-div 34.5%

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

      3. sqrt-div 34.5%

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

   6. Applied egg-rr 29.7%

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

      1. unpow2 29.7%

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

   8. Simplified 33.9%

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

   9. Taylor expanded in k around inf 29.3%

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

       1. associate-/l* 29.3%

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

   11. Simplified 29.3%

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

   if 1.69999999999999986e-235 < t < 6.20000000000000013e-115

   1. Initial program 38.1%

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

   3. Simplified 38.1%

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

      1. associate-*r* 38.1%

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

      2. add-sqr-sqrt 38.1%

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

      3. times-frac 38.1%

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

      4. div-inv 38.1%

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

      5. frac-times 38.1%

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

      6. metadata-eval 38.1%

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

   6. Applied egg-rr 38.1%

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

   8. Simplified 52.8%

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

      1. add-cube-cbrt 52.8%

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

      2. pow3 52.8%

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

   10. Applied egg-rr 88.2%

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

   11. Taylor expanded in k around inf 88.2%

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

   if 6.20000000000000013e-115 < t < 2.69999999999999983e85

   1. Initial program 65.5%

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

   3. Simplified 62.7%

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

      1. associate-*r* 68.3%

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

      2. add-sqr-sqrt 68.1%

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

      3. times-frac 68.3%

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

      4. div-inv 68.3%

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

      5. frac-times 68.3%

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

      6. metadata-eval 68.3%

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

   6. Applied egg-rr 68.3%

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

   8. Simplified 85.2%

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

      1. frac-times 90.1%

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

   10. Applied egg-rr 90.1%

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

   if 2.69999999999999983e85 < t

   1. Initial program 46.3%

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

   3. Simplified 46.3%

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

      1. associate-*r* 51.1%

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

      2. add-sqr-sqrt 51.1%

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

      3. times-frac 51.1%

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

      4. div-inv 51.1%

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

      5. frac-times 51.1%

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

      6. metadata-eval 51.1%

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

   6. Applied egg-rr 51.1%

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

   8. Simplified 51.1%

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

      1. add-cube-cbrt 51.1%

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

      2. pow3 51.1%

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

   10. Applied egg-rr 79.5%

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

       1. cbrt-div 98.7%

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

   12. Applied egg-rr 98.7%

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

   13. Taylor expanded in k around 0 93.4%

       \[\leadsto {\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}{t} \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \sqrt{0.5}}}\right)}^{2}\right)}^{3} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-235}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-115}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\frac{t \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \frac{2}{\tan k}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}{t} \cdot {\left(\sqrt[3]{\ell \cdot \sqrt{0.5}}\right)}^{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{t\_2}}{\sqrt[3]{\sin k}}}{t\_m} \cdot {\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{\ell}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{t\_2}{\sin k}}}{t\_m}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (tan k))))
   (*
    t_s
    (if (<= k 1.05e-154)
      (pow
       (*
        (/ (/ (cbrt t_2) (cbrt (sin k))) t_m)
        (pow (* (pow 0.5 0.16666666666666666) (cbrt l)) 2.0))
       3.0)
      (pow
       (*
        (pow (cbrt (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))) 2.0)
        (/ (cbrt (/ t_2 (sin k))) t_m))
       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 t_2 = 2.0 / tan(k);
	double tmp;
	if (k <= 1.05e-154) {
		tmp = pow((((cbrt(t_2) / cbrt(sin(k))) / t_m) * pow((pow(0.5, 0.16666666666666666) * cbrt(l)), 2.0)), 3.0);
	} else {
		tmp = pow((pow(cbrt((l / hypot(1.0, hypot(1.0, (k / t_m))))), 2.0) * (cbrt((t_2 / sin(k))) / t_m)), 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 t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (k <= 1.05e-154) {
		tmp = Math.pow((((Math.cbrt(t_2) / Math.cbrt(Math.sin(k))) / t_m) * Math.pow((Math.pow(0.5, 0.16666666666666666) * Math.cbrt(l)), 2.0)), 3.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt((l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m))))), 2.0) * (Math.cbrt((t_2 / Math.sin(k))) / t_m)), 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)
	t_2 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (k <= 1.05e-154)
		tmp = Float64(Float64(Float64(cbrt(t_2) / cbrt(sin(k))) / t_m) * (Float64((0.5 ^ 0.16666666666666666) * cbrt(l)) ^ 2.0)) ^ 3.0;
	else
		tmp = Float64((cbrt(Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))) ^ 2.0) * Float64(cbrt(Float64(t_2 / sin(k))) / t_m)) ^ 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_] := Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.05e-154], N[Power[N[(N[(N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[Power[0.5, 0.16666666666666666], $MachinePrecision] * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(N[Power[N[Power[N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(t$95$2 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.04999999999999992e-154

   1. Initial program 58.4%

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

   3. Simplified 59.5%

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

      1. associate-*r* 63.8%

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

      2. add-sqr-sqrt 63.7%

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

      3. times-frac 63.8%

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

      4. div-inv 63.8%

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

      5. frac-times 63.8%

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

      6. metadata-eval 63.8%

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

   6. Applied egg-rr 63.8%

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

   8. Simplified 67.6%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

   10. Applied egg-rr 76.5%

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

       1. cbrt-div 94.3%

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

   12. Applied egg-rr 94.3%

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

   13. Taylor expanded in k around 0 82.5%

       \[\leadsto {\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}{t} \cdot {\color{blue}{\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{\ell}\right)}}^{2}\right)}^{3} \]

   if 1.04999999999999992e-154 < k

   1. Initial program 59.4%

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

   3. Simplified 57.1%

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

      1. associate-*r* 60.6%

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

      2. add-sqr-sqrt 60.5%

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

      3. times-frac 60.5%

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

      4. div-inv 60.5%

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

      5. frac-times 60.5%

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

      6. metadata-eval 60.5%

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

   6. Applied egg-rr 60.5%

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

   8. Simplified 73.4%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}{t} \cdot {\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{\ell}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot \frac{t\_2}{\cos k}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t\_2 \cdot \left(t\_m \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\frac{t\_m \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0)))
   (*
    t_s
    (if (<= k 2.8e-155)
      (/
       2.0
       (*
        (* (tan k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (if (<= k 7.8e+24)
        (/
         2.0
         (pow
          (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (/ t_2 (cos k)))))
          3.0))
        (if (<= k 2.45e+135)
          (* 2.0 (* (pow l 2.0) (/ (cos k) (* t_2 (* t_m (pow k 2.0))))))
          (pow
           (*
            (/ (cbrt (/ (/ 2.0 (tan k)) (sin k))) t_m)
            (pow (cbrt (/ (* t_m l) k)) 2.0))
           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 t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 2.8e-155) {
		tmp = 2.0 / ((tan(k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else if (k <= 7.8e+24) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * (t_2 / cos(k))))), 3.0);
	} else if (k <= 2.45e+135) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (t_2 * (t_m * pow(k, 2.0)))));
	} else {
		tmp = pow(((cbrt(((2.0 / tan(k)) / sin(k))) / t_m) * pow(cbrt(((t_m * l) / k)), 2.0)), 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 t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 2.8e-155) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else if (k <= 7.8e+24) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * (t_2 / Math.cos(k))))), 3.0);
	} else if (k <= 2.45e+135) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (t_2 * (t_m * Math.pow(k, 2.0)))));
	} else {
		tmp = Math.pow(((Math.cbrt(((2.0 / Math.tan(k)) / Math.sin(k))) / t_m) * Math.pow(Math.cbrt(((t_m * l) / k)), 2.0)), 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)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 2.8e-155)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (k <= 7.8e+24)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * Float64(t_2 / cos(k))))) ^ 3.0));
	elseif (k <= 2.45e+135)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(t_2 * Float64(t_m * (k ^ 2.0))))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(2.0 / tan(k)) / sin(k))) / t_m) * (cbrt(Float64(Float64(t_m * l) / k)) ^ 2.0)) ^ 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_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-155], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.8e+24], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.45e+135], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 2.8e-155

   1. Initial program 58.4%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. associate-/r* 32.2%

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

      4. *-commutative 32.2%

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

      5. sqrt-prod 9.2%

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

      6. associate-/r* 7.8%

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

      7. sqrt-div 7.8%

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

      8. sqrt-pow1 10.4%

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

      9. metadata-eval 10.4%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 14.1%

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

   4. Applied egg-rr 14.1%

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

      1. *-commutative 14.1%

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

   6. Simplified 14.1%

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

   if 2.8e-155 < k < 7.7999999999999995e24

   1. Initial program 55.4%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in t around inf 74.4%

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

      1. add-cube-cbrt 74.2%

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

      2. pow3 74.1%

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

      3. cbrt-prod 74.1%

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

      4. associate-/l/ 72.1%

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

      5. pow2 72.1%

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

      6. cbrt-div 72.1%

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

      7. unpow3 72.1%

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

      8. add-cbrt-cube 81.5%

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

      9. pow2 81.5%

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

      10. cbrt-unprod 89.8%

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

      11. pow2 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 7.7999999999999995e24 < k < 2.4500000000000001e135

   1. Initial program 76.7%

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

   3. Simplified 76.6%

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

      1. associate-*r* 82.8%

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

      2. add-sqr-sqrt 82.8%

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

      3. times-frac 82.7%

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

      4. div-inv 82.7%

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

      5. frac-times 82.7%

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

      6. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in k around inf 82.3%

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

       1. associate-/l* 82.3%

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

       2. associate-*r* 82.3%

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

   11. Simplified 82.3%

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

   if 2.4500000000000001e135 < k

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

      1. associate-*r* 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.0%

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

      4. div-inv 57.0%

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

      5. frac-times 57.0%

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

      6. metadata-eval 57.0%

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

   6. Applied egg-rr 57.0%

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

   8. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

   10. Applied egg-rr 92.2%

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

   11. Taylor expanded in k around inf 84.7%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot \frac{{\sin k}^{2}}{\cos k}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\frac{t \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;{\left(t\_2 \cdot {\left(\sqrt[3]{\ell \cdot \sqrt{0.5}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_2 \cdot {\left(\sqrt[3]{\frac{t\_m \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (/ (cbrt (/ (/ 2.0 (tan k)) (sin k))) t_m)))
   (*
    t_s
    (if (<= k 1.05e-154)
      (/
       2.0
       (*
        (* (tan k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (if (<= k 1.3e+24)
        (pow (* t_2 (pow (cbrt (* l (sqrt 0.5))) 2.0)) 3.0)
        (if (<= k 1.5e+138)
          (*
           2.0
           (*
            (pow l 2.0)
            (/ (cos k) (* (pow (sin k) 2.0) (* t_m (pow k 2.0))))))
          (pow (* t_2 (pow (cbrt (/ (* t_m l) k)) 2.0)) 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 t_2 = cbrt(((2.0 / tan(k)) / sin(k))) / t_m;
	double tmp;
	if (k <= 1.05e-154) {
		tmp = 2.0 / ((tan(k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else if (k <= 1.3e+24) {
		tmp = pow((t_2 * pow(cbrt((l * sqrt(0.5))), 2.0)), 3.0);
	} else if (k <= 1.5e+138) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (pow(sin(k), 2.0) * (t_m * pow(k, 2.0)))));
	} else {
		tmp = pow((t_2 * pow(cbrt(((t_m * l) / k)), 2.0)), 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 t_2 = Math.cbrt(((2.0 / Math.tan(k)) / Math.sin(k))) / t_m;
	double tmp;
	if (k <= 1.05e-154) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else if (k <= 1.3e+24) {
		tmp = Math.pow((t_2 * Math.pow(Math.cbrt((l * Math.sqrt(0.5))), 2.0)), 3.0);
	} else if (k <= 1.5e+138) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * (t_m * Math.pow(k, 2.0)))));
	} else {
		tmp = Math.pow((t_2 * Math.pow(Math.cbrt(((t_m * l) / k)), 2.0)), 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)
	t_2 = Float64(cbrt(Float64(Float64(2.0 / tan(k)) / sin(k))) / t_m)
	tmp = 0.0
	if (k <= 1.05e-154)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (k <= 1.3e+24)
		tmp = Float64(t_2 * (cbrt(Float64(l * sqrt(0.5))) ^ 2.0)) ^ 3.0;
	elseif (k <= 1.5e+138)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * Float64(t_m * (k ^ 2.0))))));
	else
		tmp = Float64(t_2 * (cbrt(Float64(Float64(t_m * l) / k)) ^ 2.0)) ^ 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_] := Block[{t$95$2 = N[(N[Power[N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.05e-154], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.3e+24], N[Power[N[(t$95$2 * N[Power[N[Power[N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k, 1.5e+138], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$2 * N[Power[N[Power[N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 1.04999999999999992e-154

   1. Initial program 58.4%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. associate-/r* 32.2%

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

      4. *-commutative 32.2%

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

      5. sqrt-prod 9.2%

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

      6. associate-/r* 7.8%

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

      7. sqrt-div 7.8%

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

      8. sqrt-pow1 10.4%

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

      9. metadata-eval 10.4%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 14.1%

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

   4. Applied egg-rr 14.1%

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

      1. *-commutative 14.1%

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

   6. Simplified 14.1%

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

   if 1.04999999999999992e-154 < k < 1.2999999999999999e24

   1. Initial program 55.4%

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

   3. Simplified 49.8%

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

      1. associate-*r* 55.0%

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

      2. add-sqr-sqrt 54.7%

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

      3. times-frac 54.8%

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

      4. div-inv 54.8%

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

      5. frac-times 54.8%

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

      6. metadata-eval 54.8%

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

   6. Applied egg-rr 54.8%

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

   8. Simplified 71.9%

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

      1. add-cube-cbrt 71.7%

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

      2. pow3 71.7%

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

   10. Applied egg-rr 94.3%

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

   11. Taylor expanded in k around 0 89.6%

       \[\leadsto {\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \sqrt{0.5}}}\right)}^{2}\right)}^{3} \]

   if 1.2999999999999999e24 < k < 1.50000000000000005e138

   1. Initial program 76.7%

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

   3. Simplified 76.6%

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

      1. associate-*r* 82.8%

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

      2. add-sqr-sqrt 82.8%

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

      3. times-frac 82.7%

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

      4. div-inv 82.7%

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

      5. frac-times 82.7%

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

      6. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in k around inf 82.3%

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

       1. associate-/l* 82.3%

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

       2. associate-*r* 82.3%

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

   11. Simplified 82.3%

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

   if 1.50000000000000005e138 < k

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

      1. associate-*r* 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.0%

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

      4. div-inv 57.0%

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

      5. frac-times 57.0%

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

      6. metadata-eval 57.0%

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

   6. Applied egg-rr 57.0%

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

   8. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

   10. Applied egg-rr 92.2%

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

   11. Taylor expanded in k around inf 84.7%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\ell \cdot \sqrt{0.5}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\frac{t \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\frac{t\_m \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \]

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 5.2e-157)
    (/
     2.0
     (*
      (* (tan k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0))
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
    (if (<= k 1.95e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (if (<= k 4.6e+139)
        (/
         2.0
         (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k)))) (pow l 2.0)))
        (pow
         (*
          (/ (cbrt (/ (/ 2.0 (tan k)) (sin k))) t_m)
          (pow (cbrt (/ (* t_m l) k)) 2.0))
         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 (k <= 5.2e-157) {
		tmp = 2.0 / ((tan(k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else if (k <= 1.95e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else if (k <= 4.6e+139) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k)))) / pow(l, 2.0));
	} else {
		tmp = pow(((cbrt(((2.0 / tan(k)) / sin(k))) / t_m) * pow(cbrt(((t_m * l) / k)), 2.0)), 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 (k <= 5.2e-157) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else if (k <= 1.95e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else if (k <= 4.6e+139) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / Math.pow(l, 2.0));
	} else {
		tmp = Math.pow(((Math.cbrt(((2.0 / Math.tan(k)) / Math.sin(k))) / t_m) * Math.pow(Math.cbrt(((t_m * l) / k)), 2.0)), 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 (k <= 5.2e-157)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (k <= 1.95e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	elseif (k <= 4.6e+139)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k)))) / (l ^ 2.0)));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(2.0 / tan(k)) / sin(k))) / t_m) * (cbrt(Float64(Float64(t_m * l) / k)) ^ 2.0)) ^ 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[k, 5.2e-157], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.95e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.6e+139], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 5.19999999999999977e-157

   1. Initial program 58.4%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. associate-/r* 32.2%

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

      4. *-commutative 32.2%

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

      5. sqrt-prod 9.2%

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

      6. associate-/r* 7.8%

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

      7. sqrt-div 7.8%

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

      8. sqrt-pow1 10.4%

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

      9. metadata-eval 10.4%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 14.1%

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

   4. Applied egg-rr 14.1%

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

      1. *-commutative 14.1%

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

   6. Simplified 14.1%

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

   if 5.19999999999999977e-157 < k < 1.94999999999999989e-16

   1. Initial program 55.1%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 69.2%

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

      1. add-cube-cbrt 68.9%

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

      2. pow3 68.9%

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

      3. cbrt-prod 68.8%

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

      4. associate-/l/ 69.4%

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

      5. pow2 69.4%

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

      6. cbrt-div 69.3%

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

      7. unpow3 69.3%

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

      8. add-cbrt-cube 80.7%

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

      9. pow2 80.7%

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

      10. cbrt-unprod 87.8%

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

      11. pow2 87.8%

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

   7. Applied egg-rr 87.8%

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

   if 1.94999999999999989e-16 < k < 4.6e139

   1. Initial program 70.9%

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

   3. Simplified 79.4%

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

      1. add-sqr-sqrt 25.2%

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

      2. pow2 25.2%

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

      3. associate-/r* 20.8%

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

      4. sqrt-div 20.8%

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

      5. sqrt-pow1 20.8%

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

      6. metadata-eval 20.8%

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

      7. sqrt-prod 8.3%

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

      8. add-sqr-sqrt 25.2%

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

   6. Applied egg-rr 25.2%

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

   7. Taylor expanded in t around 0 79.5%

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

      1. times-frac 76.2%

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

   9. Simplified 76.2%

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

       1. associate-*l/ 79.5%

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

       2. associate-/l* 79.5%

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

   11. Applied egg-rr 79.5%

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

   if 4.6e139 < k

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

      1. associate-*r* 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.0%

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

      4. div-inv 57.0%

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

      5. frac-times 57.0%

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

      6. metadata-eval 57.0%

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

   6. Applied egg-rr 57.0%

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

   8. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

   10. Applied egg-rr 92.2%

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

   11. Taylor expanded in k around inf 84.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\frac{t \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{k}}}{t\_m \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t\_m} \cdot {\left(\sqrt[3]{\frac{t\_m \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \end{array} \]

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 2e-194)
    (/
     (* (pow (/ (cbrt (/ 2.0 k)) (* t_m (cbrt (sin k)))) 3.0) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 1.7e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (if (<= k 3e+136)
        (/
         2.0
         (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k)))) (pow l 2.0)))
        (pow
         (*
          (/ (cbrt (/ (/ 2.0 (tan k)) (sin k))) t_m)
          (pow (cbrt (/ (* t_m l) k)) 2.0))
         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 (k <= 2e-194) {
		tmp = (pow((cbrt((2.0 / k)) / (t_m * cbrt(sin(k)))), 3.0) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.7e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else if (k <= 3e+136) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k)))) / pow(l, 2.0));
	} else {
		tmp = pow(((cbrt(((2.0 / tan(k)) / sin(k))) / t_m) * pow(cbrt(((t_m * l) / k)), 2.0)), 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 (k <= 2e-194) {
		tmp = (Math.pow((Math.cbrt((2.0 / k)) / (t_m * Math.cbrt(Math.sin(k)))), 3.0) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.7e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else if (k <= 3e+136) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / Math.pow(l, 2.0));
	} else {
		tmp = Math.pow(((Math.cbrt(((2.0 / Math.tan(k)) / Math.sin(k))) / t_m) * Math.pow(Math.cbrt(((t_m * l) / k)), 2.0)), 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 (k <= 2e-194)
		tmp = Float64(Float64((Float64(cbrt(Float64(2.0 / k)) / Float64(t_m * cbrt(sin(k)))) ^ 3.0) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.7e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	elseif (k <= 3e+136)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k)))) / (l ^ 2.0)));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(2.0 / tan(k)) / sin(k))) / t_m) * (cbrt(Float64(Float64(t_m * l) / k)) ^ 2.0)) ^ 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[k, 2e-194], N[(N[(N[Power[N[(N[Power[N[(2.0 / k), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3e+136], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 2.00000000000000004e-194

   1. Initial program 58.3%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in k around 0 57.0%

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

      1. add-cube-cbrt 56.9%

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

      2. pow3 56.9%

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

      3. cbrt-div 56.9%

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

      4. *-commutative 56.9%

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

      5. cbrt-prod 56.9%

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

      6. unpow3 56.9%

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

      7. add-cbrt-cube 67.0%

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

   7. Applied egg-rr 67.0%

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

   if 2.00000000000000004e-194 < k < 1.7e-16

   1. Initial program 56.2%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 65.0%

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

      5. pow2 65.0%

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

      6. cbrt-div 64.9%

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

      7. unpow3 64.9%

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

      8. add-cbrt-cube 76.0%

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

      9. pow2 76.0%

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

      10. cbrt-unprod 84.0%

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

      11. pow2 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 1.7e-16 < k < 2.99999999999999979e136

   1. Initial program 70.9%

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

   3. Simplified 79.4%

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

      1. add-sqr-sqrt 25.2%

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

      2. pow2 25.2%

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

      3. associate-/r* 20.8%

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

      4. sqrt-div 20.8%

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

      5. sqrt-pow1 20.8%

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

      6. metadata-eval 20.8%

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

      7. sqrt-prod 8.3%

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

      8. add-sqr-sqrt 25.2%

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

   6. Applied egg-rr 25.2%

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

   7. Taylor expanded in t around 0 79.5%

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

      1. times-frac 76.2%

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

   9. Simplified 76.2%

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

       1. associate-*l/ 79.5%

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

       2. associate-/l* 79.5%

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

   11. Applied egg-rr 79.5%

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

   if 2.99999999999999979e136 < k

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

      1. associate-*r* 56.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. times-frac 57.0%

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

      4. div-inv 57.0%

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

      5. frac-times 57.0%

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

      6. metadata-eval 57.0%

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

   6. Applied egg-rr 57.0%

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

   8. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

   10. Applied egg-rr 92.2%

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

   11. Taylor expanded in k around inf 84.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{k}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\frac{2}{\tan k}}{\sin k}}}{t} \cdot {\left(\sqrt[3]{\frac{t \cdot \ell}{k}}\right)}^{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{k}}}{t\_m \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-194)
    (/
     (* (pow (/ (cbrt (/ 2.0 k)) (* t_m (cbrt (sin k)))) 3.0) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 8.2e-17)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (/
       2.0
       (/
        (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k))))
        (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-194) {
		tmp = (pow((cbrt((2.0 / k)) / (t_m * cbrt(sin(k)))), 3.0) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 8.2e-17) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k)))) / pow(l, 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 (k <= 2e-194) {
		tmp = (Math.pow((Math.cbrt((2.0 / k)) / (t_m * Math.cbrt(Math.sin(k)))), 3.0) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 8.2e-17) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / Math.pow(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 (k <= 2e-194)
		tmp = Float64(Float64((Float64(cbrt(Float64(2.0 / k)) / Float64(t_m * cbrt(sin(k)))) ^ 3.0) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 8.2e-17)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k)))) / (l ^ 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[k, 2e-194], N[(N[(N[Power[N[(N[Power[N[(2.0 / k), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.2e-17], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.00000000000000004e-194

   1. Initial program 58.3%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in k around 0 57.0%

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

      1. add-cube-cbrt 56.9%

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

      2. pow3 56.9%

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

      3. cbrt-div 56.9%

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

      4. *-commutative 56.9%

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

      5. cbrt-prod 56.9%

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

      6. unpow3 56.9%

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

      7. add-cbrt-cube 67.0%

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

   7. Applied egg-rr 67.0%

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

   if 2.00000000000000004e-194 < k < 8.2000000000000001e-17

   1. Initial program 56.2%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 65.0%

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

      5. pow2 65.0%

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

      6. cbrt-div 64.9%

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

      7. unpow3 64.9%

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

      8. add-cbrt-cube 76.0%

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

      9. pow2 76.0%

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

      10. cbrt-unprod 84.0%

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

      11. pow2 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 8.2000000000000001e-17 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

       1. associate-*l/ 77.2%

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

       2. associate-/l* 77.2%

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

   11. Applied egg-rr 77.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{k}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-169}:\\ \;\;\;\;{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t\_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e-169)
    (pow
     (* (* l (* (sqrt 0.5) (/ (sqrt 2.0) k))) (sqrt (/ 1.0 (pow t_m 3.0))))
     2.0)
    (if (<= k 1.95e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (/
       2.0
       (/
        (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k))))
        (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-169) {
		tmp = pow(((l * (sqrt(0.5) * (sqrt(2.0) / k))) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.95e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k)))) / pow(l, 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 (k <= 2.4e-169) {
		tmp = Math.pow(((l * (Math.sqrt(0.5) * (Math.sqrt(2.0) / k))) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.95e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / Math.pow(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 (k <= 2.4e-169)
		tmp = Float64(Float64(l * Float64(sqrt(0.5) * Float64(sqrt(2.0) / k))) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 1.95e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k)))) / (l ^ 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[k, 2.4e-169], N[Power[N[(N[(l * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.95e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.40000000000000011e-169

   1. Initial program 59.3%

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

   3. Simplified 60.4%

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

      1. add-sqr-sqrt 38.8%

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

      2. sqrt-div 37.5%

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

      3. sqrt-div 37.5%

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

   6. Applied egg-rr 36.0%

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

      1. unpow2 36.0%

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

   8. Simplified 38.6%

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

   9. Taylor expanded in k around 0 39.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 39.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 39.1%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 39.1%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 2.40000000000000011e-169 < k < 1.94999999999999989e-16

   1. Initial program 51.9%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 67.3%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 64.7%

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

      5. pow2 64.7%

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

      6. cbrt-div 64.7%

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

      7. unpow3 64.7%

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

      8. add-cbrt-cube 77.4%

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

      9. pow2 77.4%

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

      10. cbrt-unprod 86.6%

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

      11. pow2 86.6%

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

   7. Applied egg-rr 86.6%

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

   if 1.94999999999999989e-16 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

       1. associate-*l/ 77.2%

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

       2. associate-/l* 77.2%

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

   11. Applied egg-rr 77.2%

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

Alternative 10: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t\_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-170)
    (pow
     (* (* l (* (sqrt 0.5) (/ (sqrt 2.0) k))) (sqrt (/ 1.0 (pow t_m 3.0))))
     2.0)
    (if (<= k 1.25e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (cos k) (* t_m (pow (sin k) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-170) {
		tmp = pow(((l * (sqrt(0.5) * (sqrt(2.0) / k))) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.25e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

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 <= 6.5e-170) {
		tmp = Math.pow(((l * (Math.sqrt(0.5) * (Math.sqrt(2.0) / k))) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.25e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-170)
		tmp = Float64(Float64(l * Float64(sqrt(0.5) * Float64(sqrt(2.0) / k))) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 1.25e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

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, 6.5e-170], N[Power[N[(N[(l * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.25e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.50000000000000035e-170

   1. Initial program 59.3%

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

   3. Simplified 60.4%

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

      1. add-sqr-sqrt 38.8%

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

      2. sqrt-div 37.5%

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

      3. sqrt-div 37.5%

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

   6. Applied egg-rr 36.0%

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

      1. unpow2 36.0%

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

   8. Simplified 38.6%

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

   9. Taylor expanded in k around 0 39.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 39.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 39.1%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 39.1%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 6.50000000000000035e-170 < k < 1.2500000000000001e-16

   1. Initial program 51.9%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 67.3%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 64.7%

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

      5. pow2 64.7%

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

      6. cbrt-div 64.7%

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

      7. unpow3 64.7%

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

      8. add-cbrt-cube 77.4%

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

      9. pow2 77.4%

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

      10. cbrt-unprod 86.6%

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

      11. pow2 86.6%

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

   7. Applied egg-rr 86.6%

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

   if 1.2500000000000001e-16 < k

   1. Initial program 61.7%

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

   3. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.1%

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

   5. Simplified 76.1%

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

Alternative 11: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-169}:\\ \;\;\;\;{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t\_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e-169)
    (pow
     (* (* l (* (sqrt 0.5) (/ (sqrt 2.0) k))) (sqrt (/ 1.0 (pow t_m 3.0))))
     2.0)
    (if (<= k 1.6e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (/
       2.0
       (* (/ (* k k) (pow l 2.0)) (/ (* t_m (pow (sin k) 2.0)) (cos 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 (k <= 2.4e-169) {
		tmp = pow(((l * (sqrt(0.5) * (sqrt(2.0) / k))) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.6e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * pow(sin(k), 2.0)) / cos(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 (k <= 2.4e-169) {
		tmp = Math.pow(((l * (Math.sqrt(0.5) * (Math.sqrt(2.0) / k))) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.6e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(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 (k <= 2.4e-169)
		tmp = Float64(Float64(l * Float64(sqrt(0.5) * Float64(sqrt(2.0) / k))) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 1.6e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(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[k, 2.4e-169], N[Power[N[(N[(l * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.6e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.40000000000000011e-169

   1. Initial program 59.3%

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

   3. Simplified 60.4%

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

      1. add-sqr-sqrt 38.8%

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

      2. sqrt-div 37.5%

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

      3. sqrt-div 37.5%

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

   6. Applied egg-rr 36.0%

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

      1. unpow2 36.0%

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

   8. Simplified 38.6%

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

   9. Taylor expanded in k around 0 39.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 39.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 39.1%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 39.1%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 2.40000000000000011e-169 < k < 1.60000000000000011e-16

   1. Initial program 51.9%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 67.3%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 64.7%

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

      5. pow2 64.7%

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

      6. cbrt-div 64.7%

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

      7. unpow3 64.7%

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

      8. add-cbrt-cube 77.4%

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

      9. pow2 77.4%

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

      10. cbrt-unprod 86.6%

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

      11. pow2 86.6%

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

   7. Applied egg-rr 86.6%

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

   if 1.60000000000000011e-16 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

       1. unpow2 76.0%

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

   11. Applied egg-rr 76.0%

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

Alternative 12: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.75e-194)
    (/
     (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0)))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 1.85e-16)
      (/
       2.0
       (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (/
       2.0
       (* (/ (* k k) (pow l 2.0)) (/ (* t_m (pow (sin k) 2.0)) (cos 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 (k <= 1.75e-194) {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.85e-16) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * pow(sin(k), 2.0)) / cos(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 (k <= 1.75e-194) {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.85e-16) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(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 (k <= 1.75e-194)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.85e-16)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(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[k, 1.75e-194], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.85e-16], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.7500000000000001e-194

   1. Initial program 58.3%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in k around 0 57.0%

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

      1. add-cube-cbrt 56.9%

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

      2. pow2 56.9%

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

      3. cbrt-div 57.0%

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

      4. *-commutative 57.0%

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

      5. cbrt-prod 56.9%

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

      6. unpow3 56.9%

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

      7. add-cbrt-cube 56.9%

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

      8. cbrt-div 56.9%

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

      9. *-commutative 56.9%

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

      10. cbrt-prod 56.9%

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

      11. unpow3 56.9%

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

      12. add-cbrt-cube 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. unpow2 67.0%

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

      2. unpow3 67.0%

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

      3. cube-div 62.2%

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

      4. rem-cube-cbrt 62.2%

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

      5. *-commutative 62.2%

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

   9. Simplified 62.2%

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

   if 1.7500000000000001e-194 < k < 1.85e-16

   1. Initial program 56.2%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 66.9%

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

      4. associate-/l/ 65.0%

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

      5. pow2 65.0%

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

      6. cbrt-div 64.9%

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

      7. unpow3 64.9%

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

      8. add-cbrt-cube 76.0%

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

      9. pow2 76.0%

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

      10. cbrt-unprod 84.0%

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

      11. pow2 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 1.85e-16 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

       1. unpow2 76.0%

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

   11. Applied egg-rr 76.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot t\_2}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t\_2 \cdot {t\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t\_m}{k}\right)\\ \mathbf{elif}\;t\_m \leq 9.6 \cdot 10^{+95}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t\_m}^{3}}\right) \cdot \frac{\ell}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0)) (t_3 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-156)
      (/ 2.0 (* (/ (* k k) (pow l 2.0)) (/ (* t_m t_2) (cos k))))
      (if (<= t_m 5e-38)
        (*
         (* 2.0 (* (/ l k) (/ (cos k) (* t_2 (pow t_m 2.0)))))
         (* l (/ t_m k)))
        (if (<= t_m 9.6e+95)
          (* (* l (/ (/ 2.0 (tan k)) (* (sin k) (pow t_m 3.0)))) (/ l t_3))
          (/
           (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0)))
           t_3)))))))

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(sin(k), 2.0);
	double t_3 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * t_2) / cos(k)));
	} else if (t_m <= 5e-38) {
		tmp = (2.0 * ((l / k) * (cos(k) / (t_2 * pow(t_m, 2.0))))) * (l * (t_m / k));
	} else if (t_m <= 9.6e+95) {
		tmp = (l * ((2.0 / tan(k)) / (sin(k) * pow(t_m, 3.0)))) * (l / t_3);
	} else {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / t_3;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * t_2) / Math.cos(k)));
	} else if (t_m <= 5e-38) {
		tmp = (2.0 * ((l / k) * (Math.cos(k) / (t_2 * Math.pow(t_m, 2.0))))) * (l * (t_m / k));
	} else if (t_m <= 9.6e+95) {
		tmp = (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow(t_m, 3.0)))) * (l / t_3);
	} else {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / t_3;
	}
	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 = sin(k) ^ 2.0
	t_3 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.6e-156)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * t_2) / cos(k))));
	elseif (t_m <= 5e-38)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(cos(k) / Float64(t_2 * (t_m ^ 2.0))))) * Float64(l * Float64(t_m / k)));
	elseif (t_m <= 9.6e+95)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (t_m ^ 3.0)))) * Float64(l / t_3));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / t_3);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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-156], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e-38], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.6e+95], N[(N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999991e-156

   1. Initial program 61.9%

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

   3. Simplified 63.2%

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

      1. add-sqr-sqrt 7.1%

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

      2. pow2 7.1%

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

      3. associate-/r* 6.5%

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

      4. sqrt-div 6.5%

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

      5. sqrt-pow1 7.1%

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

      6. metadata-eval 7.1%

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

      7. sqrt-prod 1.9%

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

      8. add-sqr-sqrt 7.7%

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

   6. Applied egg-rr 7.7%

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

   7. Taylor expanded in t around 0 70.5%

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

      1. times-frac 70.3%

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

   9. Simplified 70.3%

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

       1. unpow2 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1.59999999999999991e-156 < t < 5.00000000000000033e-38

   1. Initial program 54.2%

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

   3. Simplified 50.3%

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

      1. associate-*r* 54.9%

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

      2. add-sqr-sqrt 54.9%

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

      3. times-frac 55.1%

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

      4. div-inv 55.1%

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

      5. frac-times 55.1%

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

      6. metadata-eval 55.1%

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

   6. Applied egg-rr 55.1%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in k around inf 42.8%

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

       1. times-frac 42.7%

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

   11. Simplified 42.7%

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

   12. Taylor expanded in k around inf 85.4%

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

       1. associate-/l* 85.4%

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

   14. Simplified 85.4%

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

   if 5.00000000000000033e-38 < t < 9.6000000000000002e95

   1. Initial program 76.7%

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

   3. Simplified 76.1%

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

      1. associate-*r* 81.1%

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

      2. *-un-lft-identity 81.1%

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

      3. times-frac 81.1%

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

      4. div-inv 81.2%

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

      5. frac-times 81.3%

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

      6. metadata-eval 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. /-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.1%

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

      4. *-commutative 81.1%

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

   8. Simplified 81.1%

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

   if 9.6000000000000002e95 < t

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in k around 0 45.3%

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

      1. add-cube-cbrt 45.3%

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

      2. pow2 45.3%

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

      3. cbrt-div 45.3%

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

      4. *-commutative 45.3%

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

      5. cbrt-prod 45.3%

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

      6. unpow3 45.3%

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

      7. add-cbrt-cube 45.3%

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

      8. cbrt-div 45.3%

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

      9. *-commutative 45.3%

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

      10. cbrt-prod 45.3%

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

      11. unpow3 45.3%

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

      12. add-cbrt-cube 63.5%

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

   7. Applied egg-rr 63.5%

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

      1. unpow2 63.5%

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

      2. unpow3 63.5%

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

      3. cube-div 56.2%

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

      4. rem-cube-cbrt 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t}{k}\right)\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+95}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot t\_2}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.02 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t\_2 \cdot {t\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t\_m}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t\_m}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 1.6e-156)
      (/ 2.0 (* (/ (* k k) (pow l 2.0)) (/ (* t_m t_2) (cos k))))
      (if (<= t_m 1.02e-38)
        (*
         (* 2.0 (* (/ l k) (/ (cos k) (* t_2 (pow t_m 2.0)))))
         (* l (/ t_m k)))
        (*
         (* l (/ (/ 2.0 (tan k)) (* (sin k) (pow t_m 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 t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * t_2) / cos(k)));
	} else if (t_m <= 1.02e-38) {
		tmp = (2.0 * ((l / k) * (cos(k) / (t_2 * pow(t_m, 2.0))))) * (l * (t_m / k));
	} else {
		tmp = (l * ((2.0 / tan(k)) / (sin(k) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (t_m <= 1.6d-156) then
        tmp = 2.0d0 / (((k * k) / (l ** 2.0d0)) * ((t_m * t_2) / cos(k)))
    else if (t_m <= 1.02d-38) then
        tmp = (2.0d0 * ((l / k) * (cos(k) / (t_2 * (t_m ** 2.0d0))))) * (l * (t_m / k))
    else
        tmp = (l * ((2.0d0 / tan(k)) / (sin(k) * (t_m ** 3.0d0)))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * t_2) / Math.cos(k)));
	} else if (t_m <= 1.02e-38) {
		tmp = (2.0 * ((l / k) * (Math.cos(k) / (t_2 * Math.pow(t_m, 2.0))))) * (l * (t_m / k));
	} else {
		tmp = (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 1.6e-156:
		tmp = 2.0 / (((k * k) / math.pow(l, 2.0)) * ((t_m * t_2) / math.cos(k)))
	elif t_m <= 1.02e-38:
		tmp = (2.0 * ((l / k) * (math.cos(k) / (t_2 * math.pow(t_m, 2.0))))) * (l * (t_m / k))
	else:
		tmp = (l * ((2.0 / math.tan(k)) / (math.sin(k) * math.pow(t_m, 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)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.6e-156)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * t_2) / cos(k))));
	elseif (t_m <= 1.02e-38)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(cos(k) / Float64(t_2 * (t_m ^ 2.0))))) * Float64(l * Float64(t_m / k)));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.6e-156)
		tmp = 2.0 / (((k * k) / (l ^ 2.0)) * ((t_m * t_2) / cos(k)));
	elseif (t_m <= 1.02e-38)
		tmp = (2.0 * ((l / k) * (cos(k) / (t_2 * (t_m ^ 2.0))))) * (l * (t_m / k));
	else
		tmp = (l * ((2.0 / tan(k)) / (sin(k) * (t_m ^ 3.0)))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-156], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.02e-38], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if t < 1.59999999999999991e-156

   1. Initial program 61.9%

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

   3. Simplified 63.2%

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

      1. add-sqr-sqrt 7.1%

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

      2. pow2 7.1%

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

      3. associate-/r* 6.5%

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

      4. sqrt-div 6.5%

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

      5. sqrt-pow1 7.1%

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

      6. metadata-eval 7.1%

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

      7. sqrt-prod 1.9%

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

      8. add-sqr-sqrt 7.7%

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

   6. Applied egg-rr 7.7%

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

   7. Taylor expanded in t around 0 70.5%

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

      1. times-frac 70.3%

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

   9. Simplified 70.3%

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

       1. unpow2 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1.59999999999999991e-156 < t < 1.01999999999999998e-38

   1. Initial program 54.2%

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

   3. Simplified 50.3%

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

      1. associate-*r* 54.9%

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

      2. add-sqr-sqrt 54.9%

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

      3. times-frac 55.1%

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

      4. div-inv 55.1%

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

      5. frac-times 55.1%

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

      6. metadata-eval 55.1%

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

   6. Applied egg-rr 55.1%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in k around inf 42.8%

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

       1. times-frac 42.7%

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

   11. Simplified 42.7%

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

   12. Taylor expanded in k around inf 85.4%

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

       1. associate-/l* 85.4%

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

   14. Simplified 85.4%

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

   if 1.01999999999999998e-38 < t

   1. Initial program 54.1%

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

   3. Simplified 53.9%

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

      1. associate-*r* 58.9%

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

      2. *-un-lft-identity 58.9%

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

      3. times-frac 58.9%

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

      4. div-inv 58.9%

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

      5. frac-times 58.9%

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

      6. metadata-eval 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. /-rgt-identity 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-/r* 58.9%

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

      4. *-commutative 58.9%

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

   8. Simplified 58.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot t\_2}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{-12}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t\_2 \cdot {t\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t\_m}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{t\_2}{\cos k}\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 1.6e-156)
      (/ 2.0 (* (/ (* k k) (pow l 2.0)) (/ (* t_m t_2) (cos k))))
      (if (<= t_m 6.1e-12)
        (*
         (* 2.0 (* (/ l k) (/ (cos k) (* t_2 (pow t_m 2.0)))))
         (* l (/ t_m k)))
        (/
         2.0
         (* (* 2.0 (/ t_2 (cos k))) (* (/ (pow t_m 2.0) l) (/ t_m l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * t_2) / cos(k)));
	} else if (t_m <= 6.1e-12) {
		tmp = (2.0 * ((l / k) * (cos(k) / (t_2 * pow(t_m, 2.0))))) * (l * (t_m / k));
	} else {
		tmp = 2.0 / ((2.0 * (t_2 / cos(k))) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (t_m <= 1.6d-156) then
        tmp = 2.0d0 / (((k * k) / (l ** 2.0d0)) * ((t_m * t_2) / cos(k)))
    else if (t_m <= 6.1d-12) then
        tmp = (2.0d0 * ((l / k) * (cos(k) / (t_2 * (t_m ** 2.0d0))))) * (l * (t_m / k))
    else
        tmp = 2.0d0 / ((2.0d0 * (t_2 / cos(k))) * (((t_m ** 2.0d0) / l) * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 1.6e-156) {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * t_2) / Math.cos(k)));
	} else if (t_m <= 6.1e-12) {
		tmp = (2.0 * ((l / k) * (Math.cos(k) / (t_2 * Math.pow(t_m, 2.0))))) * (l * (t_m / k));
	} else {
		tmp = 2.0 / ((2.0 * (t_2 / Math.cos(k))) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 1.6e-156:
		tmp = 2.0 / (((k * k) / math.pow(l, 2.0)) * ((t_m * t_2) / math.cos(k)))
	elif t_m <= 6.1e-12:
		tmp = (2.0 * ((l / k) * (math.cos(k) / (t_2 * math.pow(t_m, 2.0))))) * (l * (t_m / k))
	else:
		tmp = 2.0 / ((2.0 * (t_2 / math.cos(k))) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.6e-156)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * t_2) / cos(k))));
	elseif (t_m <= 6.1e-12)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(cos(k) / Float64(t_2 * (t_m ^ 2.0))))) * Float64(l * Float64(t_m / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(t_2 / cos(k))) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.6e-156)
		tmp = 2.0 / (((k * k) / (l ^ 2.0)) * ((t_m * t_2) / cos(k)));
	elseif (t_m <= 6.1e-12)
		tmp = (2.0 * ((l / k) * (cos(k) / (t_2 * (t_m ^ 2.0))))) * (l * (t_m / k));
	else
		tmp = 2.0 / ((2.0 * (t_2 / cos(k))) * (((t_m ^ 2.0) / l) * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-156], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e-12], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.59999999999999991e-156

   1. Initial program 61.9%

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

   3. Simplified 63.2%

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

      1. add-sqr-sqrt 7.1%

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

      2. pow2 7.1%

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

      3. associate-/r* 6.5%

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

      4. sqrt-div 6.5%

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

      5. sqrt-pow1 7.1%

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

      6. metadata-eval 7.1%

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

      7. sqrt-prod 1.9%

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

      8. add-sqr-sqrt 7.7%

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

   6. Applied egg-rr 7.7%

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

   7. Taylor expanded in t around 0 70.5%

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

      1. times-frac 70.3%

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

   9. Simplified 70.3%

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

       1. unpow2 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1.59999999999999991e-156 < t < 6.1000000000000003e-12

   1. Initial program 58.9%

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

   3. Simplified 55.4%

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

      1. associate-*r* 59.6%

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

      2. add-sqr-sqrt 59.6%

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

      3. times-frac 59.8%

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

      4. div-inv 59.8%

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

      5. frac-times 59.8%

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

      6. metadata-eval 59.8%

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

   6. Applied egg-rr 59.8%

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

   8. Simplified 76.6%

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

   9. Taylor expanded in k around inf 45.5%

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

       1. times-frac 45.4%

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

   11. Simplified 45.4%

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

   12. Taylor expanded in k around inf 86.8%

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

       1. associate-/l* 86.8%

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

   14. Simplified 86.8%

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

   if 6.1000000000000003e-12 < t

   1. Initial program 52.2%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in t around inf 45.4%

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

      1. associate-/r* 41.9%

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

      2. unpow3 41.9%

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

      3. times-frac 55.0%

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

      4. pow2 55.0%

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

   7. Applied egg-rr 55.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-12}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right) \cdot \left(\ell \cdot \frac{t}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

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 7.6e-17)
    (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k k))))
    (/
     2.0
     (* (/ (* k k) (pow l 2.0)) (/ (* t_m (pow (sin k) 2.0)) (cos 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 (k <= 7.6e-17) {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 / (((k * k) / pow(l, 2.0)) * ((t_m * pow(sin(k), 2.0)) / cos(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 (k <= 7.6e-17) {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 / (((k * k) / Math.pow(l, 2.0)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(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 (k <= 7.6e-17)
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / (l ^ 2.0)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(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[k, 7.6e-17], 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], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.6000000000000002e-17

   1. Initial program 57.8%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. add-cube-cbrt 56.8%

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

      2. pow3 56.8%

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

      3. cbrt-prod 56.8%

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

      4. associate-/l/ 53.5%

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

      5. pow2 53.5%

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

      6. cbrt-div 54.0%

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

      7. unpow3 54.0%

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

      8. add-cbrt-cube 64.7%

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

      9. pow2 64.7%

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

      10. cbrt-unprod 69.2%

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

      11. pow2 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. cube-prod 62.5%

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

      2. rem-cube-cbrt 62.5%

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

   9. Simplified 62.5%

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

       1. unpow2 57.0%

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

   11. Applied egg-rr 62.5%

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

   if 7.6000000000000002e-17 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

       1. unpow2 76.0%

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

   11. Applied egg-rr 76.0%

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

Alternative 17: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t\_m \cdot {k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

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 8.5e-17)
    (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k k))))
    (/ 2.0 (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t_m (pow k 2.0)) (cos 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 (k <= 8.5e-17) {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * ((t_m * pow(k, 2.0)) / cos(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 (k <= 8.5e-17) {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t_m * Math.pow(k, 2.0)) / Math.cos(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 (k <= 8.5e-17)
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t_m * (k ^ 2.0)) / cos(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[k, 8.5e-17], 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], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.5e-17

   1. Initial program 57.8%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. add-cube-cbrt 56.8%

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

      2. pow3 56.8%

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

      3. cbrt-prod 56.8%

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

      4. associate-/l/ 53.5%

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

      5. pow2 53.5%

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

      6. cbrt-div 54.0%

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

      7. unpow3 54.0%

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

      8. add-cbrt-cube 64.7%

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

      9. pow2 64.7%

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

      10. cbrt-unprod 69.2%

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

      11. pow2 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. cube-prod 62.5%

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

      2. rem-cube-cbrt 62.5%

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

   9. Simplified 62.5%

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

       1. unpow2 57.0%

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

   11. Applied egg-rr 62.5%

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

   if 8.5e-17 < k

   1. Initial program 61.7%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 25.1%

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

      2. pow2 25.1%

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

      3. associate-/r* 23.2%

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

      4. sqrt-div 23.2%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

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

      7. sqrt-prod 7.8%

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

      8. add-sqr-sqrt 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in t around 0 77.2%

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

      1. times-frac 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in k around 0 70.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t\_m \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{{t\_m}^{3} \cdot \frac{\frac{t\_2}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(\frac{t\_m}{{\ell}^{0.6666666666666666}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 2.0))))
   (*
    t_s
    (if (<= t_m 7.8e-125)
      (/ (* 2.0 (pow l 2.0)) (* (pow k 3.0) (* t_m (sin k))))
      (if (<= t_m 1.15e+95)
        (/ 2.0 (* (pow t_m 3.0) (/ (/ t_2 l) l)))
        (/ 2.0 (* t_2 (pow (/ t_m (pow l 0.6666666666666666)) 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 t_2 = 2.0 * pow(k, 2.0);
	double tmp;
	if (t_m <= 7.8e-125) {
		tmp = (2.0 * pow(l, 2.0)) / (pow(k, 3.0) * (t_m * sin(k)));
	} else if (t_m <= 1.15e+95) {
		tmp = 2.0 / (pow(t_m, 3.0) * ((t_2 / l) / l));
	} else {
		tmp = 2.0 / (t_2 * pow((t_m / pow(l, 0.6666666666666666)), 3.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (k ** 2.0d0)
    if (t_m <= 7.8d-125) then
        tmp = (2.0d0 * (l ** 2.0d0)) / ((k ** 3.0d0) * (t_m * sin(k)))
    else if (t_m <= 1.15d+95) then
        tmp = 2.0d0 / ((t_m ** 3.0d0) * ((t_2 / l) / l))
    else
        tmp = 2.0d0 / (t_2 * ((t_m / (l ** 0.6666666666666666d0)) ** 3.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * Math.pow(k, 2.0);
	double tmp;
	if (t_m <= 7.8e-125) {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(k, 3.0) * (t_m * Math.sin(k)));
	} else if (t_m <= 1.15e+95) {
		tmp = 2.0 / (Math.pow(t_m, 3.0) * ((t_2 / l) / l));
	} else {
		tmp = 2.0 / (t_2 * Math.pow((t_m / Math.pow(l, 0.6666666666666666)), 3.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 * math.pow(k, 2.0)
	tmp = 0
	if t_m <= 7.8e-125:
		tmp = (2.0 * math.pow(l, 2.0)) / (math.pow(k, 3.0) * (t_m * math.sin(k)))
	elif t_m <= 1.15e+95:
		tmp = 2.0 / (math.pow(t_m, 3.0) * ((t_2 / l) / l))
	else:
		tmp = 2.0 / (t_2 * math.pow((t_m / math.pow(l, 0.6666666666666666)), 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)
	t_2 = Float64(2.0 * (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.8e-125)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((k ^ 3.0) * Float64(t_m * sin(k))));
	elseif (t_m <= 1.15e+95)
		tmp = Float64(2.0 / Float64((t_m ^ 3.0) * Float64(Float64(t_2 / l) / l)));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(t_m / (l ^ 0.6666666666666666)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 * (k ^ 2.0);
	tmp = 0.0;
	if (t_m <= 7.8e-125)
		tmp = (2.0 * (l ^ 2.0)) / ((k ^ 3.0) * (t_m * sin(k)));
	elseif (t_m <= 1.15e+95)
		tmp = 2.0 / ((t_m ^ 3.0) * ((t_2 / l) / l));
	else
		tmp = 2.0 / (t_2 * ((t_m / (l ^ 0.6666666666666666)) ^ 3.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-125], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 3.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+95], N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[(t$95$2 / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(t$95$m / N[Power[l, 0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.79999999999999965e-125

   1. Initial program 61.4%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in k around 0 60.0%

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

   6. Taylor expanded in k around inf 65.7%

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

      1. associate-*r/ 65.7%

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

   8. Simplified 65.7%

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

   if 7.79999999999999965e-125 < t < 1.14999999999999999e95

   1. Initial program 66.4%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in k around 0 54.9%

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

      1. associate-*l/ 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. associate-/l* 61.2%

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

   9. Simplified 61.2%

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

       1. associate-*l/ 59.2%

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

       2. associate-/l* 59.2%

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

   11. Applied egg-rr 59.2%

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

       1. associate-/l* 61.4%

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

       2. associate-*r/ 61.4%

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

   13. Simplified 61.4%

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

   if 1.14999999999999999e95 < t

   1. Initial program 45.3%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in k around 0 42.1%

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

      1. add-cube-cbrt 42.1%

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

      2. pow3 42.1%

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

      3. cbrt-prod 42.1%

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

      4. associate-/l/ 37.5%

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

      5. pow2 37.5%

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

      6. cbrt-div 37.5%

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

      7. unpow3 37.5%

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

      8. add-cbrt-cube 56.2%

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

      9. pow2 56.2%

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

      10. cbrt-unprod 65.6%

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

      11. pow2 65.6%

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

   7. Applied egg-rr 65.6%

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

      1. cube-prod 56.6%

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

      2. rem-cube-cbrt 56.6%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   9. Simplified 56.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. add-exp-log 56.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{e^{\log \left({\left(\sqrt[3]{\ell}\right)}^{2}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       2. log-pow 20.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{e^{\color{blue}{2 \cdot \log \left(\sqrt[3]{\ell}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

   11. Applied egg-rr 20.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{e^{2 \cdot \log \left(\sqrt[3]{\ell}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 20.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{e^{\color{blue}{\log \left(\sqrt[3]{\ell}\right) \cdot 2}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       2. exp-to-pow 56.6%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       3. pow2 56.6%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       4. pow1/3 20.9%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\ell}^{0.3333333333333333}} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       5. pow1/3 20.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\ell}^{0.3333333333333333} \cdot \color{blue}{{\ell}^{0.3333333333333333}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       6. pow-prod-up 20.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\ell}^{\left(0.3333333333333333 + 0.3333333333333333\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       7. metadata-eval 20.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\ell}^{\color{blue}{0.6666666666666666}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

   13. Applied egg-rr 20.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\ell}^{0.6666666666666666}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{{t}^{3} \cdot \frac{\frac{2 \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t}{{\ell}^{0.6666666666666666}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.95e-16)
    (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k k))))
    (/ (* 2.0 (pow l 2.0)) (* (pow k 3.0) (* t_m (sin 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 (k <= 1.95e-16) {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (pow(k, 3.0) * (t_m * sin(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 (k <= 1.95e-16) {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(k, 3.0) * (t_m * Math.sin(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 (k <= 1.95e-16)
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((k ^ 3.0) * Float64(t_m * sin(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[k, 1.95e-16], 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], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 3.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999989e-16

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\right) \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}}} \]

      2. pow3 56.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\right)}^{3}}} \]

      3. cbrt-prod 56.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}}^{3}} \]

      4. associate-/l/ 53.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      5. pow2 53.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\frac{{t}^{3}}{\color{blue}{{\ell}^{2}}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      6. cbrt-div 54.0%

         \[\leadsto \frac{2}{{\left(\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{{\ell}^{2}}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      7. unpow3 54.0%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}{\sqrt[3]{{\ell}^{2}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      8. add-cbrt-cube 64.7%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{t}}{\sqrt[3]{{\ell}^{2}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      9. pow2 64.7%

         \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\color{blue}{\ell \cdot \ell}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      10. cbrt-unprod 69.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

      11. pow2 69.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. cube-prod 62.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}} \]

      2. rem-cube-cbrt 62.5%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   9. Simplified 62.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. unpow2 57.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied egg-rr 62.5%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 1.94999999999999989e-16 < k

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 57.3%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around inf 69.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.8%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.95e-16)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (pow (/ (pow t_m 1.5) l) 2.0)))
    (/ (* 2.0 (pow l 2.0)) (* (pow k 3.0) (* t_m (sin 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 (k <= 1.95e-16) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * pow((pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (pow(k, 3.0) * (t_m * sin(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 (k <= 1.95d-16) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / ((k ** 3.0d0) * (t_m * sin(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 (k <= 1.95e-16) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(k, 3.0) * (t_m * Math.sin(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 k <= 1.95e-16:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * math.pow((math.pow(t_m, 1.5) / l), 2.0))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (math.pow(k, 3.0) * (t_m * math.sin(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 (k <= 1.95e-16)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * (Float64((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((k ^ 3.0) * Float64(t_m * sin(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 (k <= 1.95e-16)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 1.5) / l) ^ 2.0));
	else
		tmp = (2.0 * (l ^ 2.0)) / ((k ^ 3.0) * (t_m * sin(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[k, 1.95e-16], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 3.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999989e-16

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 24.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 24.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/r* 22.7%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 22.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 26.9%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 26.9%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-prod 13.2%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 30.6%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Applied egg-rr 28.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   if 1.94999999999999989e-16 < k

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 57.3%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around inf 69.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.8%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t\_m \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 7.8e-125)
    (/ (* 2.0 (pow l 2.0)) (* (pow k 3.0) (* t_m (sin k))))
    (if (<= t_m 1.35e+151)
      (/ 2.0 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (* k k)) l)))
      (/
       (* (* l l) (/ (/ 2.0 k) (* k (pow t_m 3.0))))
       (+ 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 <= 7.8e-125) {
		tmp = (2.0 * pow(l, 2.0)) / (pow(k, 3.0) * (t_m * sin(k)));
	} else if (t_m <= 1.35e+151) {
		tmp = 2.0 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = ((l * l) * ((2.0 / k) / (k * pow(t_m, 3.0)))) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.8d-125) then
        tmp = (2.0d0 * (l ** 2.0d0)) / ((k ** 3.0d0) * (t_m * sin(k)))
    else if (t_m <= 1.35d+151) then
        tmp = 2.0d0 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k * k)) / l))
    else
        tmp = ((l * l) * ((2.0d0 / k) / (k * (t_m ** 3.0d0)))) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-125) {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(k, 3.0) * (t_m * Math.sin(k)));
	} else if (t_m <= 1.35e+151) {
		tmp = 2.0 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = ((l * l) * ((2.0 / k) / (k * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.8e-125:
		tmp = (2.0 * math.pow(l, 2.0)) / (math.pow(k, 3.0) * (t_m * math.sin(k)))
	elif t_m <= 1.35e+151:
		tmp = 2.0 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l))
	else:
		tmp = ((l * l) * ((2.0 / k) / (k * math.pow(t_m, 3.0)))) / (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 <= 7.8e-125)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((k ^ 3.0) * Float64(t_m * sin(k))));
	elseif (t_m <= 1.35e+151)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * Float64(k * k)) / l)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.8e-125)
		tmp = (2.0 * (l ^ 2.0)) / ((k ^ 3.0) * (t_m * sin(k)));
	elseif (t_m <= 1.35e+151)
		tmp = 2.0 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k * k)) / l));
	else
		tmp = ((l * l) * ((2.0 / k) / (k * (t_m ^ 3.0)))) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-125], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 3.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+151], N[(2.0 / N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.79999999999999965e-125

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around inf 65.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.7%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]

   if 7.79999999999999965e-125 < t < 1.3500000000000001e151

   1. Initial program 58.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 48.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 50.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 50.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 53.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 53.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. unpow2 52.3%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied egg-rr 53.1%

       \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]
   12. Step-by-step derivation

       1. cube-mult 53.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       2. *-un-lft-identity 53.1%

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       3. times-frac 61.5%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       4. pow2 61.5%

          \[\leadsto \frac{2}{\left(\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   13. Applied egg-rr 61.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   if 1.3500000000000001e151 < t

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 48.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 48.1%

      \[\leadsto \frac{\frac{\frac{2}{k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 2e-94)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (if (<= t_m 3.9e+151)
      (/ 2.0 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (* k k)) l)))
      (/
       (* (* l l) (/ (/ 2.0 k) (* k (pow t_m 3.0))))
       (+ 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 <= 2e-94) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else if (t_m <= 3.9e+151) {
		tmp = 2.0 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = ((l * l) * ((2.0 / k) / (k * pow(t_m, 3.0)))) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-94) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else if (t_m <= 3.9d+151) then
        tmp = 2.0d0 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k * k)) / l))
    else
        tmp = ((l * l) * ((2.0d0 / k) / (k * (t_m ** 3.0d0)))) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-94) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else if (t_m <= 3.9e+151) {
		tmp = 2.0 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = ((l * l) * ((2.0 / k) / (k * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-94:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	elif t_m <= 3.9e+151:
		tmp = 2.0 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l))
	else:
		tmp = ((l * l) * ((2.0 / k) / (k * math.pow(t_m, 3.0)))) / (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 <= 2e-94)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	elseif (t_m <= 3.9e+151)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * Float64(k * k)) / l)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-94)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	elseif (t_m <= 3.9e+151)
		tmp = 2.0 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k * k)) / l));
	else
		tmp = ((l * l) * ((2.0 / k) / (k * (t_m ^ 3.0)))) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-94], 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, 3.9e+151], N[(2.0 / N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.9999999999999999e-94

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 9.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 9.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/r* 9.0%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 9.0%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 10.8%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 10.8%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-prod 3.6%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 11.5%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 11.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in t around 0 70.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. times-frac 70.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   9. Simplified 70.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   10. Taylor expanded in k around 0 63.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 63.8%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 63.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 1.9999999999999999e-94 < t < 3.89999999999999976e151

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 47.6%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 49.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 49.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 52.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 52.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied egg-rr 52.4%

       \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]
   12. Step-by-step derivation

       1. cube-mult 52.4%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       2. *-un-lft-identity 52.4%

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       3. times-frac 61.6%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       4. pow2 61.6%

          \[\leadsto \frac{2}{\left(\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   13. Applied egg-rr 61.6%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   if 3.89999999999999976e151 < t

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 48.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 48.1%

      \[\leadsto \frac{\frac{\frac{2}{k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.5e+91)
    (/ 2.0 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (* k k)) l)))
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e+91) {
		tmp = 2.0 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(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 (k <= 3.5d+91) then
        tmp = 2.0d0 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k * k)) / l))
    else
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (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 (k <= 3.5e+91) {
		tmp = 2.0 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l));
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(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 k <= 3.5e+91:
		tmp = 2.0 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / l))
	else:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(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 (k <= 3.5e+91)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * Float64(k * k)) / l)));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (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 (k <= 3.5e+91)
		tmp = 2.0 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k * k)) / l));
	else
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (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[k, 3.5e+91], N[(2.0 / N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.50000000000000001e91

   1. Initial program 58.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. unpow2 58.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied egg-rr 59.7%

       \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]
   12. Step-by-step derivation

       1. cube-mult 59.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       2. *-un-lft-identity 59.7%

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       3. times-frac 63.5%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

       4. pow2 63.5%

          \[\leadsto \frac{2}{\left(\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   13. Applied egg-rr 63.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

   if 3.50000000000000001e91 < k

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 23.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/r* 23.1%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 23.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 25.2%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 25.2%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-prod 8.5%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 29.5%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 29.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in t around 0 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. times-frac 75.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   9. Simplified 75.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   10. Taylor expanded in k around 0 67.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 67.3%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 67.3%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 57.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \end{array} \]

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 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (* k k)) 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 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / 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 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k * k)) / 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 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / 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 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * (k * k)) / 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(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * Float64(k * k)) / 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 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k * k)) / 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[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 58.3%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*l/ 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

7. Applied egg-rr 59.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
8. Step-by-step derivation

   1. associate-/l* 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

9. Simplified 58.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
10. Step-by-step derivation

    1. unpow2 61.8%

       \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

11. Applied egg-rr 58.8%

    \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]
12. Step-by-step derivation

    1. cube-mult 58.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

    2. *-un-lft-identity 58.9%

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

    3. times-frac 62.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

    4. pow2 62.8%

       \[\leadsto \frac{2}{\left(\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

13. Applied egg-rr 62.8%

    \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}\right)} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

14. Final simplification 62.8%

    \[\leadsto \frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]
15. Add Preprocessing

Alternative 25: 55.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{2 \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m \cdot {t\_m}^{2}}{\ell}} \end{array} \]

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)) l) (/ (* t_m (pow t_m 2.0)) 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)) / l) * ((t_m * pow(t_m, 2.0)) / 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)) / l) * ((t_m * (t_m ** 2.0d0)) / 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)) / l) * ((t_m * Math.pow(t_m, 2.0)) / 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)) / l) * ((t_m * math.pow(t_m, 2.0)) / 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(Float64(2.0 * Float64(k * k)) / l) * Float64(Float64(t_m * (t_m ^ 2.0)) / 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)) / l) * ((t_m * (t_m ^ 2.0)) / 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[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 58.3%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*l/ 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

7. Applied egg-rr 59.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
8. Step-by-step derivation

   1. associate-/l* 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

9. Simplified 58.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
10. Step-by-step derivation

    1. unpow2 61.8%

       \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

11. Applied egg-rr 58.8%

    \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]
12. Step-by-step derivation

    1. unpow3 58.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

    2. pow2 58.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{2}} \cdot t}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

13. Applied egg-rr 58.9%

    \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{2} \cdot t}}{\ell} \cdot \frac{2 \cdot \left(k \cdot k\right)}{\ell}} \]

14. Final simplification 58.9%

    \[\leadsto \frac{2}{\frac{2 \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t \cdot {t}^{2}}{\ell}} \]
15. Add Preprocessing

Alternative 26: 55.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{2 \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}} \end{array} \]

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)) l) (/ (pow t_m 3.0) 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)) / l) * (pow(t_m, 3.0) / 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)) / l) * ((t_m ** 3.0d0) / 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)) / l) * (Math.pow(t_m, 3.0) / 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)) / l) * (math.pow(t_m, 3.0) / 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(Float64(2.0 * Float64(k * k)) / l) * Float64((t_m ^ 3.0) / 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)) / l) * ((t_m ^ 3.0) / 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[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 58.3%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*l/ 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

7. Applied egg-rr 59.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
8. Step-by-step derivation

   1. associate-/l* 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

9. Simplified 58.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
10. Step-by-step derivation

    1. unpow2 61.8%

       \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

11. Applied egg-rr 58.8%

    \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]

12. Final simplification 58.8%

    \[\leadsto \frac{2}{\frac{2 \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))