Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 90.9%

Time: 18.2s

Alternatives: 18

Speedup: 1.9×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k\_m} \cdot \sqrt[3]{{\left(\sqrt{\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\right)}^{2}}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.3e-56)
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (*
        (cbrt (sin k_m))
        (cbrt (pow (sqrt (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0)))) 2.0))))
      3.0))
    (if (<= k_m 4e+68)
      (/
       2.0
       (*
        (* (sin k_m) (tan k_m))
        (pow (* (/ (pow t_m 1.5) l) (hypot 1.0 (hypot 1.0 (/ k_m t_m)))) 2.0)))
      (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.3e-56) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k_m)) * cbrt(pow(sqrt((tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))), 2.0)))), 3.0);
	} else if (k_m <= 4e+68) {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * pow(((pow(t_m, 1.5) / l) * hypot(1.0, hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.3e-56) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k_m)) * Math.cbrt(Math.pow(Math.sqrt((Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))), 2.0)))), 3.0);
	} else if (k_m <= 4e+68) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.3e-56)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k_m)) * cbrt((sqrt(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))) ^ 2.0)))) ^ 3.0));
	elseif (k_m <= 4e+68)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(Float64((t_m ^ 1.5) / l) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.3e-56], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[Sqrt[N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+68], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.29999999999999984e-56

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

      1. associate-*l* 49.6%

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

      2. associate-/r* 57.2%

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

      3. associate-+r+ 57.2%

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

      4. metadata-eval 57.2%

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

      5. associate-*l* 57.1%

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

      6. add-cube-cbrt 57.1%

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

      7. pow3 57.1%

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

   6. Applied egg-rr 74.9%

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

      1. pow1/3 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}}\right)}^{3}} \]

      2. associate-*l* 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}^{0.3333333333333333}\right)}^{3}} \]

      3. unpow-prod-down 37.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left({\sin k}^{0.3333333333333333} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

      4. pow1/3 45.9%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\color{blue}{\sqrt[3]{\sin k}} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   8. Applied egg-rr 45.9%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 85.5%

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

   10. Simplified 85.5%

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

       1. add-sqr-sqrt 40.5%

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

       2. pow2 40.5%

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

   12. Applied egg-rr 40.5%

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

   if 3.29999999999999984e-56 < k < 3.99999999999999981e68

   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 59.9%

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

   6. Applied egg-rr 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. associate-*r* 44.4%

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

      3. unpow-prod-down 44.2%

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

      4. pow2 44.2%

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

      5. add-sqr-sqrt 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. *-lft-identity 62.8%

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

      2. *-commutative 62.8%

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

   10. Simplified 62.8%

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

   if 3.99999999999999981e68 < k

   1. Initial program 35.6%

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

   3. Simplified 35.6%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/l* 44.3%

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

   9. Simplified 44.3%

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

       1. unpow-prod-down 40.9%

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

       2. pow2 40.9%

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

       3. add-sqr-sqrt 89.0%

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

   11. Applied egg-rr 89.0%

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

Alternative 2: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k\_m} \cdot \sqrt[3]{\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 1.85 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.6e-57)
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (sin k_m)) (cbrt (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0))))))
      3.0))
    (if (<= k_m 1.85e+68)
      (/
       2.0
       (*
        (* (sin k_m) (tan k_m))
        (pow (* (/ (pow t_m 1.5) l) (hypot 1.0 (hypot 1.0 (/ k_m t_m)))) 2.0)))
      (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6e-57) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k_m)) * cbrt((tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))))), 3.0);
	} else if (k_m <= 1.85e+68) {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * pow(((pow(t_m, 1.5) / l) * hypot(1.0, hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6e-57) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k_m)) * Math.cbrt((Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))))), 3.0);
	} else if (k_m <= 1.85e+68) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.6e-57)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k_m)) * cbrt(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))))) ^ 3.0));
	elseif (k_m <= 1.85e+68)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(Float64((t_m ^ 1.5) / l) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.6e-57], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.85e+68], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.5999999999999995e-57

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

      1. associate-*l* 49.6%

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

      2. associate-/r* 57.2%

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

      3. associate-+r+ 57.2%

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

      4. metadata-eval 57.2%

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

      5. associate-*l* 57.1%

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

      6. add-cube-cbrt 57.1%

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

      7. pow3 57.1%

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

   6. Applied egg-rr 74.9%

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

      1. pow1/3 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}}\right)}^{3}} \]

      2. associate-*l* 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}^{0.3333333333333333}\right)}^{3}} \]

      3. unpow-prod-down 37.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left({\sin k}^{0.3333333333333333} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

      4. pow1/3 45.9%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\color{blue}{\sqrt[3]{\sin k}} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   8. Applied egg-rr 45.9%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 85.5%

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

   10. Simplified 85.5%

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

   if 7.5999999999999995e-57 < k < 1.84999999999999999e68

   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 59.9%

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

   6. Applied egg-rr 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. associate-*r* 44.4%

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

      3. unpow-prod-down 44.2%

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

      4. pow2 44.2%

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

      5. add-sqr-sqrt 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. *-lft-identity 62.8%

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

      2. *-commutative 62.8%

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

   10. Simplified 62.8%

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

   if 1.84999999999999999e68 < k

   1. Initial program 35.6%

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

   3. Simplified 35.6%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/l* 44.3%

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

   9. Simplified 44.3%

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

       1. unpow-prod-down 40.9%

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

       2. pow2 40.9%

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

       3. add-sqr-sqrt 89.0%

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

   11. Applied egg-rr 89.0%

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

Alternative 3: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k\_m} \cdot \sqrt[3]{k\_m \cdot 2}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 1.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 9.6e-57)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (sin k_m)) (cbrt (* k_m 2.0))))
      3.0))
    (if (<= k_m 1.4e+68)
      (/
       2.0
       (*
        (* (sin k_m) (tan k_m))
        (pow (* (/ (pow t_m 1.5) l) (hypot 1.0 (hypot 1.0 (/ k_m t_m)))) 2.0)))
      (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 9.6e-57) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k_m)) * cbrt((k_m * 2.0)))), 3.0);
	} else if (k_m <= 1.4e+68) {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * pow(((pow(t_m, 1.5) / l) * hypot(1.0, hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 9.6e-57) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k_m)) * Math.cbrt((k_m * 2.0)))), 3.0);
	} else if (k_m <= 1.4e+68) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)))), 2.0));
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 9.6e-57)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k_m)) * cbrt(Float64(k_m * 2.0)))) ^ 3.0));
	elseif (k_m <= 1.4e+68)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(Float64((t_m ^ 1.5) / l) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 9.6e-57], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k$95$m * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.4e+68], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 9.60000000000000025e-57

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

      1. associate-*l* 49.6%

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

      2. associate-/r* 57.2%

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

      3. associate-+r+ 57.2%

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

      4. metadata-eval 57.2%

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

      5. associate-*l* 57.1%

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

      6. add-cube-cbrt 57.1%

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

      7. pow3 57.1%

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

   6. Applied egg-rr 74.9%

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

      1. pow1/3 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}}\right)}^{3}} \]

      2. associate-*l* 67.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}^{0.3333333333333333}\right)}^{3}} \]

      3. unpow-prod-down 37.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left({\sin k}^{0.3333333333333333} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

      4. pow1/3 45.9%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\color{blue}{\sqrt[3]{\sin k}} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   8. Applied egg-rr 45.9%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 85.5%

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

   10. Simplified 85.5%

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

   11. Taylor expanded in k around 0 71.7%

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

       1. *-commutative 71.7%

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

   13. Simplified 71.7%

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

   if 9.60000000000000025e-57 < k < 1.4e68

   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 59.9%

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

   6. Applied egg-rr 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. associate-*r* 44.4%

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

      3. unpow-prod-down 44.2%

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

      4. pow2 44.2%

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

      5. add-sqr-sqrt 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. *-lft-identity 62.8%

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

      2. *-commutative 62.8%

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

   10. Simplified 62.8%

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

   if 1.4e68 < k

   1. Initial program 35.6%

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

   3. Simplified 35.6%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/l* 44.3%

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

   9. Simplified 44.3%

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

       1. unpow-prod-down 40.9%

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

       2. pow2 40.9%

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

       3. add-sqr-sqrt 89.0%

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

   11. Applied egg-rr 89.0%

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

Alternative 4: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k\_m} \cdot \sqrt[3]{k\_m \cdot 2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 8.2e-13)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (sin k_m)) (cbrt (* k_m 2.0))))
      3.0))
    (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e-13) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k_m)) * cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e-13) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k_m)) * Math.cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 8.2e-13)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k_m)) * cbrt(Float64(k_m * 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 8.2e-13], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k$95$m * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000004e-13

   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 54.2%

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

      1. associate-*l* 50.5%

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

      2. associate-/r* 57.7%

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

      3. associate-+r+ 57.7%

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

      4. metadata-eval 57.7%

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

      5. associate-*l* 57.7%

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

      6. add-cube-cbrt 57.7%

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

      7. pow3 57.7%

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

   6. Applied egg-rr 75.3%

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

      1. pow1/3 68.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}}\right)}^{3}} \]

      2. associate-*l* 68.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}^{0.3333333333333333}\right)}^{3}} \]

      3. unpow-prod-down 38.8%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left({\sin k}^{0.3333333333333333} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

      4. pow1/3 47.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\color{blue}{\sqrt[3]{\sin k}} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   8. Applied egg-rr 47.3%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 85.5%

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

   10. Simplified 85.5%

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

   11. Taylor expanded in k around 0 72.2%

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

       1. *-commutative 72.2%

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

   13. Simplified 72.2%

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

   if 8.2000000000000004e-13 < k

   1. Initial program 41.0%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 20.3%

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

   7. Taylor expanded in t around 0 41.9%

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

      1. associate-/l* 42.0%

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

   9. Simplified 42.0%

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

       1. unpow-prod-down 39.5%

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

       2. pow2 39.5%

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

       3. add-sqr-sqrt 84.3%

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

   11. Applied egg-rr 84.3%

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

Alternative 5: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k\_m \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.6)
    (/
     2.0
     (pow
      (* (/ (pow t_m 1.5) l) (* k_m (hypot 1.0 (hypot 1.0 (/ k_m t_m)))))
      2.0))
    (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))), 2.0);
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 7.6:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k_m * math.hypot(1.0, math.hypot(1.0, (k_m / t_m))))), 2.0)
	else:
		tmp = 2.0 / (math.pow((k_m * (math.sin(k_m) / l)), 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.6)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m * hypot(1.0, hypot(1.0, Float64(k_m / t_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.6)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))) ^ 2.0);
	else
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.6], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.5999999999999996

   1. Initial program 54.9%

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

   3. Simplified 55.0%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in k around 0 41.9%

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

   if 7.5999999999999996 < k

   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%

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

   6. Applied egg-rr 14.6%

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

   7. Taylor expanded in t around 0 39.1%

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

      1. associate-/l* 39.1%

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

   9. Simplified 39.1%

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

       1. unpow-prod-down 36.5%

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

       2. pow2 36.5%

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

       3. add-sqr-sqrt 84.6%

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

   11. Applied egg-rr 84.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.7 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {\left(\frac{k\_m \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.7e-14)
    (* 2.0 (pow (* (/ (* k_m (sqrt 2.0)) l) (sqrt (pow t_m 3.0))) -2.0))
    (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.7e-14) {
		tmp = 2.0 * pow((((k_m * sqrt(2.0)) / l) * sqrt(pow(t_m, 3.0))), -2.0);
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.7d-14) then
        tmp = 2.0d0 * ((((k_m * sqrt(2.0d0)) / l) * sqrt((t_m ** 3.0d0))) ** (-2.0d0))
    else
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) ** 2.0d0) * (t_m / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.7e-14) {
		tmp = 2.0 * Math.pow((((k_m * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t_m, 3.0))), -2.0);
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.7e-14:
		tmp = 2.0 * math.pow((((k_m * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t_m, 3.0))), -2.0)
	else:
		tmp = 2.0 / (math.pow((k_m * (math.sin(k_m) / l)), 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.7e-14)
		tmp = Float64(2.0 * (Float64(Float64(Float64(k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.7e-14)
		tmp = 2.0 * ((((k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ -2.0);
	else
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.7e-14], N[(2.0 * N[Power[N[(N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.7000000000000002e-14

   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 54.2%

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

   6. Applied egg-rr 34.8%

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

      1. div-inv 34.8%

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

      2. pow-flip 34.8%

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

      3. associate-*r* 34.8%

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

      4. metadata-eval 34.8%

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

   8. Applied egg-rr 34.8%

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

   9. Taylor expanded in k around 0 38.3%

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

   if 4.7000000000000002e-14 < k

   1. Initial program 41.0%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 20.3%

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

   7. Taylor expanded in t around 0 41.9%

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

      1. associate-/l* 42.0%

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

   9. Simplified 42.0%

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

       1. unpow-prod-down 39.5%

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

       2. pow2 39.5%

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

       3. add-sqr-sqrt 84.3%

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

   11. Applied egg-rr 84.3%

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

Alternative 7: 77.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-144}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{{t\_m}^{-3}}{\sin k\_m} \cdot \frac{2}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell} \cdot \left(2 \cdot {k\_m}^{2}\right)}\\ \mathbf{elif}\;k\_m \leq 7 \cdot 10^{+37}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m \cdot \tan k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e-144)
    (* l (* (* (/ (pow t_m -3.0) (sin k_m)) (/ 2.0 (tan k_m))) (* l 0.5)))
    (if (<= k_m 2.5e-86)
      (/ 2.0 (* (/ (* (pow t_m 2.0) (/ t_m l)) l) (* 2.0 (pow k_m 2.0))))
      (if (<= k_m 7e+37)
        (* (* l 0.5) (* l (/ (/ 2.0 (pow t_m 3.0)) (* (sin k_m) (tan k_m)))))
        (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-144) {
		tmp = l * (((pow(t_m, -3.0) / sin(k_m)) * (2.0 / tan(k_m))) * (l * 0.5));
	} else if (k_m <= 2.5e-86) {
		tmp = 2.0 / (((pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * pow(k_m, 2.0)));
	} else if (k_m <= 7e+37) {
		tmp = (l * 0.5) * (l * ((2.0 / pow(t_m, 3.0)) / (sin(k_m) * tan(k_m))));
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-144) then
        tmp = l * ((((t_m ** (-3.0d0)) / sin(k_m)) * (2.0d0 / tan(k_m))) * (l * 0.5d0))
    else if (k_m <= 2.5d-86) then
        tmp = 2.0d0 / ((((t_m ** 2.0d0) * (t_m / l)) / l) * (2.0d0 * (k_m ** 2.0d0)))
    else if (k_m <= 7d+37) then
        tmp = (l * 0.5d0) * (l * ((2.0d0 / (t_m ** 3.0d0)) / (sin(k_m) * tan(k_m))))
    else
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) ** 2.0d0) * (t_m / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-144) {
		tmp = l * (((Math.pow(t_m, -3.0) / Math.sin(k_m)) * (2.0 / Math.tan(k_m))) * (l * 0.5));
	} else if (k_m <= 2.5e-86) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * Math.pow(k_m, 2.0)));
	} else if (k_m <= 7e+37) {
		tmp = (l * 0.5) * (l * ((2.0 / Math.pow(t_m, 3.0)) / (Math.sin(k_m) * Math.tan(k_m))));
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e-144:
		tmp = l * (((math.pow(t_m, -3.0) / math.sin(k_m)) * (2.0 / math.tan(k_m))) * (l * 0.5))
	elif k_m <= 2.5e-86:
		tmp = 2.0 / (((math.pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * math.pow(k_m, 2.0)))
	elif k_m <= 7e+37:
		tmp = (l * 0.5) * (l * ((2.0 / math.pow(t_m, 3.0)) / (math.sin(k_m) * math.tan(k_m))))
	else:
		tmp = 2.0 / (math.pow((k_m * (math.sin(k_m) / l)), 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-144)
		tmp = Float64(l * Float64(Float64(Float64((t_m ^ -3.0) / sin(k_m)) * Float64(2.0 / tan(k_m))) * Float64(l * 0.5)));
	elseif (k_m <= 2.5e-86)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l) * Float64(2.0 * (k_m ^ 2.0))));
	elseif (k_m <= 7e+37)
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(Float64(2.0 / (t_m ^ 3.0)) / Float64(sin(k_m) * tan(k_m)))));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-144)
		tmp = l * ((((t_m ^ -3.0) / sin(k_m)) * (2.0 / tan(k_m))) * (l * 0.5));
	elseif (k_m <= 2.5e-86)
		tmp = 2.0 / ((((t_m ^ 2.0) * (t_m / l)) / l) * (2.0 * (k_m ^ 2.0)));
	elseif (k_m <= 7e+37)
		tmp = (l * 0.5) * (l * ((2.0 / (t_m ^ 3.0)) / (sin(k_m) * tan(k_m))));
	else
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e-144], N[(l * N[(N[(N[(N[Power[t$95$m, -3.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.5e-86], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7e+37], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 2.39999999999999994e-144

   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 48.2%

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

      1. div-inv 48.2%

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

      2. associate-*r* 55.3%

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

      3. associate-*l* 55.3%

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

   6. Applied egg-rr 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

      3. associate-*r/ 55.3%

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

      4. *-rgt-identity 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in k around 0 52.6%

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

       1. *-commutative 52.6%

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

   11. Simplified 52.6%

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

       1. pow1 52.6%

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

       2. associate-*l* 52.6%

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

       3. associate-/r* 57.7%

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

       4. div-inv 57.7%

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

       5. pow-flip 57.7%

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

       6. metadata-eval 57.7%

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

   13. Applied egg-rr 57.7%

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

       1. unpow1 57.7%

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

       2. associate-/l/ 52.6%

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

       3. *-commutative 52.6%

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

       4. *-commutative 52.6%

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

       5. times-frac 57.7%

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

   15. Simplified 57.7%

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

   if 2.39999999999999994e-144 < k < 2.4999999999999999e-86

   1. Initial program 79.8%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in k around 0 81.0%

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

      1. unpow3 81.0%

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

      2. *-un-lft-identity 81.0%

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

      3. times-frac 90.8%

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

      4. pow2 90.8%

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

   7. Applied egg-rr 90.8%

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

   if 2.4999999999999999e-86 < k < 7e37

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

      1. div-inv 57.8%

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

      2. associate-*r* 65.8%

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

      3. associate-*l* 65.4%

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

   6. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-/r* 65.5%

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

      3. associate-*r/ 65.5%

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

      4. *-rgt-identity 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in k around 0 79.2%

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

       1. *-commutative 79.2%

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

   11. Simplified 79.2%

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

   if 7e37 < k

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in t around 0 40.2%

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

      1. associate-/l* 40.2%

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

   9. Simplified 40.2%

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

       1. unpow-prod-down 37.3%

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

       2. pow2 37.3%

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

       3. add-sqr-sqrt 85.9%

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

   11. Applied egg-rr 85.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-144}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{{t}^{-3}}{\sin k} \cdot \frac{2}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+37}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \left(\left(\frac{{t\_m}^{-3}}{\sin k\_m} \cdot \frac{2}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{-146}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;k\_m \leq 9.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell} \cdot \left(2 \cdot {k\_m}^{2}\right)}\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2
         (*
          l
          (* (* (/ (pow t_m -3.0) (sin k_m)) (/ 2.0 (tan k_m))) (* l 0.5)))))
   (*
    t_s
    (if (<= k_m 5.5e-146)
      t_2
      (if (<= k_m 9.2e-86)
        (/ 2.0 (* (/ (* (pow t_m 2.0) (/ t_m l)) l) (* 2.0 (pow k_m 2.0))))
        (if (<= k_m 5e+34)
          t_2
          (/ 2.0 (* (pow (* k_m (/ (sin k_m) l)) 2.0) (/ t_m (cos k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l * (((pow(t_m, -3.0) / sin(k_m)) * (2.0 / tan(k_m))) * (l * 0.5));
	double tmp;
	if (k_m <= 5.5e-146) {
		tmp = t_2;
	} else if (k_m <= 9.2e-86) {
		tmp = 2.0 / (((pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * pow(k_m, 2.0)));
	} else if (k_m <= 5e+34) {
		tmp = t_2;
	} else {
		tmp = 2.0 / (pow((k_m * (sin(k_m) / l)), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * ((((t_m ** (-3.0d0)) / sin(k_m)) * (2.0d0 / tan(k_m))) * (l * 0.5d0))
    if (k_m <= 5.5d-146) then
        tmp = t_2
    else if (k_m <= 9.2d-86) then
        tmp = 2.0d0 / ((((t_m ** 2.0d0) * (t_m / l)) / l) * (2.0d0 * (k_m ** 2.0d0)))
    else if (k_m <= 5d+34) then
        tmp = t_2
    else
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) ** 2.0d0) * (t_m / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l * (((Math.pow(t_m, -3.0) / Math.sin(k_m)) * (2.0 / Math.tan(k_m))) * (l * 0.5));
	double tmp;
	if (k_m <= 5.5e-146) {
		tmp = t_2;
	} else if (k_m <= 9.2e-86) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * Math.pow(k_m, 2.0)));
	} else if (k_m <= 5e+34) {
		tmp = t_2;
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = l * (((math.pow(t_m, -3.0) / math.sin(k_m)) * (2.0 / math.tan(k_m))) * (l * 0.5))
	tmp = 0
	if k_m <= 5.5e-146:
		tmp = t_2
	elif k_m <= 9.2e-86:
		tmp = 2.0 / (((math.pow(t_m, 2.0) * (t_m / l)) / l) * (2.0 * math.pow(k_m, 2.0)))
	elif k_m <= 5e+34:
		tmp = t_2
	else:
		tmp = 2.0 / (math.pow((k_m * (math.sin(k_m) / l)), 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(l * Float64(Float64(Float64((t_m ^ -3.0) / sin(k_m)) * Float64(2.0 / tan(k_m))) * Float64(l * 0.5)))
	tmp = 0.0
	if (k_m <= 5.5e-146)
		tmp = t_2;
	elseif (k_m <= 9.2e-86)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l) * Float64(2.0 * (k_m ^ 2.0))));
	elseif (k_m <= 5e+34)
		tmp = t_2;
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = l * ((((t_m ^ -3.0) / sin(k_m)) * (2.0 / tan(k_m))) * (l * 0.5));
	tmp = 0.0;
	if (k_m <= 5.5e-146)
		tmp = t_2;
	elseif (k_m <= 9.2e-86)
		tmp = 2.0 / ((((t_m ^ 2.0) * (t_m / l)) / l) * (2.0 * (k_m ^ 2.0)));
	elseif (k_m <= 5e+34)
		tmp = t_2;
	else
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(l * N[(N[(N[(N[Power[t$95$m, -3.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 5.5e-146], t$95$2, If[LessEqual[k$95$m, 9.2e-86], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5e+34], t$95$2, N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 5.49999999999999998e-146 or 9.19999999999999985e-86 < k < 4.9999999999999998e34

   1. Initial program 52.6%

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

   3. Simplified 49.4%

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

      1. div-inv 49.4%

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

      2. associate-*r* 56.6%

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

      3. associate-*l* 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-/r* 56.6%

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

      3. associate-*r/ 56.6%

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

      4. *-rgt-identity 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in k around 0 55.9%

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

       1. *-commutative 55.9%

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

   11. Simplified 55.9%

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

       1. pow1 55.9%

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

       2. associate-*l* 55.9%

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

       3. associate-/r* 60.4%

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

       4. div-inv 60.4%

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

       5. pow-flip 60.4%

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

       6. metadata-eval 60.4%

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

   13. Applied egg-rr 60.4%

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

       1. unpow1 60.4%

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

       2. associate-/l/ 55.9%

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

       3. *-commutative 55.9%

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

       4. *-commutative 55.9%

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

       5. times-frac 60.4%

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

   15. Simplified 60.4%

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

   if 5.49999999999999998e-146 < k < 9.19999999999999985e-86

   1. Initial program 79.8%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in k around 0 81.0%

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

      1. unpow3 81.0%

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

      2. *-un-lft-identity 81.0%

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

      3. times-frac 90.8%

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

      4. pow2 90.8%

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

   7. Applied egg-rr 90.8%

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

   if 4.9999999999999998e34 < k

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in t around 0 40.2%

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

      1. associate-/l* 40.2%

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

   9. Simplified 40.2%

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

       1. unpow-prod-down 37.3%

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

       2. pow2 37.3%

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

       3. add-sqr-sqrt 85.9%

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

   11. Applied egg-rr 85.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-146}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{{t}^{-3}}{\sin k} \cdot \frac{2}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{{t}^{-3}}{\sin k} \cdot \frac{2}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \frac{\sin k\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_2}^{2} \cdot \frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (/ (sin k_m) l))))
   (*
    t_s
    (if (<= k_m 1.6e-162)
      (/ 2.0 (pow (* t_2 (sqrt t_m)) 2.0))
      (if (<= k_m 8.4e-13)
        (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t_m 3.0) l)) l))
        (/ 2.0 (* (pow t_2 2.0) (/ t_m (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * (sin(k_m) / l);
	double tmp;
	if (k_m <= 1.6e-162) {
		tmp = 2.0 / pow((t_2 * sqrt(t_m)), 2.0);
	} else if (k_m <= 8.4e-13) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / (pow(t_2, 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = k_m * (sin(k_m) / l)
    if (k_m <= 1.6d-162) then
        tmp = 2.0d0 / ((t_2 * sqrt(t_m)) ** 2.0d0)
    else if (k_m <= 8.4d-13) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((t_2 ** 2.0d0) * (t_m / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * (Math.sin(k_m) / l);
	double tmp;
	if (k_m <= 1.6e-162) {
		tmp = 2.0 / Math.pow((t_2 * Math.sqrt(t_m)), 2.0);
	} else if (k_m <= 8.4e-13) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / (Math.pow(t_2, 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = k_m * (math.sin(k_m) / l)
	tmp = 0
	if k_m <= 1.6e-162:
		tmp = 2.0 / math.pow((t_2 * math.sqrt(t_m)), 2.0)
	elif k_m <= 8.4e-13:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = 2.0 / (math.pow(t_2, 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * Float64(sin(k_m) / l))
	tmp = 0.0
	if (k_m <= 1.6e-162)
		tmp = Float64(2.0 / (Float64(t_2 * sqrt(t_m)) ^ 2.0));
	elseif (k_m <= 8.4e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64((t_2 ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = k_m * (sin(k_m) / l);
	tmp = 0.0;
	if (k_m <= 1.6e-162)
		tmp = 2.0 / ((t_2 * sqrt(t_m)) ^ 2.0);
	elseif (k_m <= 8.4e-13)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = 2.0 / ((t_2 ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.6e-162], N[(2.0 / N[Power[N[(t$95$2 * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.4e-13], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.59999999999999988e-162

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

   6. Applied egg-rr 31.7%

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

   7. Taylor expanded in t around 0 38.9%

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

      1. associate-/l* 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in k around 0 37.5%

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

   if 1.59999999999999988e-162 < k < 8.39999999999999955e-13

   1. Initial program 73.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in k around 0 78.1%

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

      1. associate-*l/ 80.7%

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

   7. Applied egg-rr 80.7%

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

   if 8.39999999999999955e-13 < k

   1. Initial program 41.0%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 20.3%

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

   7. Taylor expanded in t around 0 41.9%

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

      1. associate-/l* 42.0%

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

   9. Simplified 42.0%

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

       1. unpow-prod-down 39.5%

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

       2. pow2 39.5%

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

       3. add-sqr-sqrt 84.3%

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

   11. Applied egg-rr 84.3%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k\_m} \cdot {\left(\frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.6e-162)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt t_m)) 2.0))
    (if (<= k_m 6.5e-12)
      (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t_m 3.0) l)) l))
      (/ 2.0 (* (/ t_m (cos k_m)) (pow (/ (* k_m (sin k_m)) l) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.6e-162) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt(t_m)), 2.0);
	} else if (k_m <= 6.5e-12) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((t_m / cos(k_m)) * pow(((k_m * sin(k_m)) / l), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.6d-162) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt(t_m)) ** 2.0d0)
    else if (k_m <= 6.5d-12) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((t_m / cos(k_m)) * (((k_m * sin(k_m)) / l) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.6e-162) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt(t_m)), 2.0);
	} else if (k_m <= 6.5e-12) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k_m)) * Math.pow(((k_m * Math.sin(k_m)) / l), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.6e-162:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt(t_m)), 2.0)
	elif k_m <= 6.5e-12:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = 2.0 / ((t_m / math.cos(k_m)) * math.pow(((k_m * math.sin(k_m)) / l), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.6e-162)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(t_m)) ^ 2.0));
	elseif (k_m <= 6.5e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k_m)) * (Float64(Float64(k_m * sin(k_m)) / l) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.6e-162)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt(t_m)) ^ 2.0);
	elseif (k_m <= 6.5e-12)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = 2.0 / ((t_m / cos(k_m)) * (((k_m * sin(k_m)) / l) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.6e-162], N[(2.0 / N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.5e-12], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.59999999999999988e-162

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

   6. Applied egg-rr 31.7%

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

   7. Taylor expanded in t around 0 38.9%

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

      1. associate-/l* 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in k around 0 37.5%

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

   if 1.59999999999999988e-162 < k < 6.5000000000000002e-12

   1. Initial program 73.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in k around 0 78.1%

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

      1. associate-*l/ 80.7%

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

   7. Applied egg-rr 80.7%

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

   if 6.5000000000000002e-12 < k

   1. Initial program 41.0%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 20.3%

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

   7. Taylor expanded in t around 0 41.9%

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

      1. associate-/l* 42.0%

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

   9. Simplified 42.0%

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

   10. Taylor expanded in k around inf 67.9%

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

       1. associate-/l* 64.3%

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

       2. unpow2 64.3%

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

       3. *-commutative 64.3%

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

       4. times-frac 64.3%

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

       5. unpow2 64.3%

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

       6. unpow2 64.3%

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

       7. times-frac 64.3%

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

       8. rem-square-sqrt 27.4%

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

       9. swap-sqr 28.7%

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

       10. swap-sqr 41.7%

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

       11. associate-*r* 41.8%

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

       12. associate-*r* 42.0%

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

       13. swap-sqr 39.5%

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

   12. Simplified 84.3%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.36 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.36e-69)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt t_m)) 2.0))
    (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t_m 3.0) l)) l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.36e-69) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.36d-69) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.36e-69) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.36e-69:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.36e-69)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.36e-69)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.36e-69], N[(2.0 / N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35999999999999998e-69

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in t around 0 38.0%

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

      1. associate-/l* 38.0%

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

   9. Simplified 38.0%

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

   10. Taylor expanded in k around 0 27.4%

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

   if 1.35999999999999998e-69 < t

   1. Initial program 61.3%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. associate-*l/ 62.1%

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

   7. Applied egg-rr 62.1%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.36 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.82 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.82e-69)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t_m 3.0) l)) l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.82e-69) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.82d-69) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.82e-69) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.82e-69:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.82e-69)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.82e-69)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.82e-69], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.82e-69

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in t around 0 38.0%

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

      1. associate-/l* 38.0%

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

   9. Simplified 38.0%

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

   10. Taylor expanded in k around 0 24.7%

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

   if 1.82e-69 < t

   1. Initial program 61.3%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. associate-*l/ 62.1%

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

   7. Applied egg-rr 62.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.82 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e+24)
    (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t_m 3.0) l)) l))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e+24) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d+24) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e+24) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e+24:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e+24)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e+24)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e+24], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e24

   1. Initial program 53.8%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in k around 0 57.1%

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

      1. associate-*l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

   if 2.4000000000000001e24 < k

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. times-frac 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in k around 0 56.2%

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

   9. Taylor expanded in k around 0 56.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.6e+24)
    (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k_m 2.0)) l)))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.6e+24) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.6d+24) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.6e+24) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.6e+24:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.6e+24)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.6e+24)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.6e+24], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.59999999999999983e24

   1. Initial program 53.8%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in k around 0 57.1%

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

      1. associate-*l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. associate-/l* 58.6%

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

   9. Simplified 58.6%

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

   if 3.59999999999999983e24 < k

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. times-frac 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in k around 0 56.2%

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

   9. Taylor expanded in k around 0 56.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2e+25)
    (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (/ (pow t_m 3.0) l) l)))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2e+25) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2d+25) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2e+25) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2e+25:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t_m, 3.0) / l) / l))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2e+25)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2e+25)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t_m ^ 3.0) / l) / l));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2e+25], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000018e25

   1. Initial program 53.8%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in k around 0 57.1%

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

   if 2.00000000000000018e25 < k

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. times-frac 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in k around 0 56.2%

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

   9. Taylor expanded in k around 0 56.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{{k\_m}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{\frac{2}{{k\_m}^{2}}}{{t\_m}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.4e-72)
    (/ 2.0 (* (pow k_m 4.0) (/ t_m (pow l 2.0))))
    (* (* l 0.5) (* l (/ (/ 2.0 (pow k_m 2.0)) (pow t_m 3.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 6.4e-72) {
		tmp = 2.0 / (pow(k_m, 4.0) * (t_m / pow(l, 2.0)));
	} else {
		tmp = (l * 0.5) * (l * ((2.0 / pow(k_m, 2.0)) / pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 6.4d-72) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else
        tmp = (l * 0.5d0) * (l * ((2.0d0 / (k_m ** 2.0d0)) / (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 6.4e-72) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else {
		tmp = (l * 0.5) * (l * ((2.0 / Math.pow(k_m, 2.0)) / Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 6.4e-72:
		tmp = 2.0 / (math.pow(k_m, 4.0) * (t_m / math.pow(l, 2.0)))
	else:
		tmp = (l * 0.5) * (l * ((2.0 / math.pow(k_m, 2.0)) / math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 6.4e-72)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	else
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(Float64(2.0 / (k_m ^ 2.0)) / (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 6.4e-72)
		tmp = 2.0 / ((k_m ^ 4.0) * (t_m / (l ^ 2.0)));
	else
		tmp = (l * 0.5) * (l * ((2.0 / (k_m ^ 2.0)) / (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.4e-72], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.39999999999999998e-72

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   6. Applied egg-rr 18.3%

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

   7. Taylor expanded in t around 0 37.6%

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

      1. associate-/l* 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in k around 0 54.7%

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

       1. associate-*r/ 55.9%

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

   12. Simplified 55.9%

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

   if 6.39999999999999998e-72 < t

   1. Initial program 60.6%

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

   3. Simplified 56.4%

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

      1. div-inv 56.4%

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

      2. associate-*r* 64.1%

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

      3. associate-*l* 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.4%

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

      3. associate-*r/ 67.4%

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

      4. *-rgt-identity 67.4%

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

   8. Simplified 67.4%

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

   9. Taylor expanded in k around 0 53.9%

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

       1. *-commutative 53.9%

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

   11. Simplified 53.9%

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

   12. Taylor expanded in k around 0 57.9%

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

       1. associate-/r* 56.7%

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

   14. Simplified 56.7%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{\frac{2}{{k}^{2}}}{{t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

6. Applied egg-rr 30.6%

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

7. Taylor expanded in t around 0 39.0%

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

   1. associate-/l* 39.1%

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

9. Simplified 39.1%

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

10. Taylor expanded in k around 0 52.5%

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

    1. associate-*r/ 53.7%

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

12. Simplified 53.7%

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

Alternative 18: 51.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in t around 0 61.4%

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

   1. associate-*r* 61.4%

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

   2. times-frac 62.5%

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

7. Simplified 62.5%

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

8. Taylor expanded in k around 0 52.5%

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

9. Taylor expanded in k around 0 52.5%

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

10. Final simplification 52.5%

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

Reproduce

herbie shell --seed 2024101 
(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))))