Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 84.8%

Time: 21.1s

Alternatives: 17

Speedup: 3.8×

• Could not converge on a ground truth

  1. reg0 = (bf "3.530860191974567249956010709759045366027e-296")
  2. reg1 = (bf "4.051163519106676441621031351877208810934e-292")
  3. reg2 = (bf "3.099782099025579449116473326865537316196e-285")

• 39.5% 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: 35.6% 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: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := \frac{\sqrt{2}}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot e^{\log t\_2 + -2 \cdot \log l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{{\left(t\_m \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot \sqrt[3]{\frac{t\_2}{\cos k}}\right)\right)}^{2}} \cdot \frac{t\_3 \cdot {\left(\sqrt[3]{l\_m}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)) (t_3 (/ (sqrt 2.0) k)))
   (*
    t_s
    (if (<= t_m 1.05e-107)
      (/
       2.0
       (*
        (/ (* t_m (pow k 2.0)) (cos k))
        (exp (+ (log t_2) (* -2.0 (log l_m))))))
      (*
       (/
        (* t_m t_3)
        (pow (* t_m (* (pow (cbrt l_m) -2.0) (cbrt (/ t_2 (cos k))))) 2.0))
       (/ (* t_3 (pow (cbrt l_m) 2.0)) (cbrt (* (sin k) (tan k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(sin(k), 2.0);
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (t_m <= 1.05e-107) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * exp((log(t_2) + (-2.0 * log(l_m)))));
	} else {
		tmp = ((t_m * t_3) / pow((t_m * (pow(cbrt(l_m), -2.0) * cbrt((t_2 / cos(k))))), 2.0)) * ((t_3 * pow(cbrt(l_m), 2.0)) / cbrt((sin(k) * tan(k))));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (t_m <= 1.05e-107) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * Math.exp((Math.log(t_2) + (-2.0 * Math.log(l_m)))));
	} else {
		tmp = ((t_m * t_3) / Math.pow((t_m * (Math.pow(Math.cbrt(l_m), -2.0) * Math.cbrt((t_2 / Math.cos(k))))), 2.0)) * ((t_3 * Math.pow(Math.cbrt(l_m), 2.0)) / Math.cbrt((Math.sin(k) * Math.tan(k))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (t_m <= 1.05e-107)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * exp(Float64(log(t_2) + Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(t_m * t_3) / (Float64(t_m * Float64((cbrt(l_m) ^ -2.0) * cbrt(Float64(t_2 / cos(k))))) ^ 2.0)) * Float64(Float64(t_3 * (cbrt(l_m) ^ 2.0)) / cbrt(Float64(sin(k) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-107], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[t$95$2], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-107

   1. Initial program 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. pow2 78.0%

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

      2. add-log-exp 59.9%

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

      3. div-inv 59.8%

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

      4. exp-prod 64.1%

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

      5. pow2 64.1%

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

      6. pow-flip 64.1%

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

      7. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. pow-exp 59.8%

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

       2. rem-log-exp 78.0%

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

       3. pow-to-exp 35.6%

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

       4. pow-to-exp 17.3%

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

       5. prod-exp 18.9%

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

       6. rem-log-exp 18.9%

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

       7. pow-to-exp 44.3%

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

       8. rem-log-exp 40.9%

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

       9. pow-to-exp 76.6%

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

       10. log-pow 44.3%

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

   11. Applied egg-rr 44.3%

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

   if 1.05e-107 < t

   1. Initial program 26.2%

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

   3. Simplified 26.2%

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

   6. Applied egg-rr 88.4%

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

      1. associate-/r/ 88.4%

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

      2. associate-/r* 88.4%

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

      3. associate-/r/ 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 87.5%

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

   10. Applied egg-rr 87.5%

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

       1. associate-/l* 88.4%

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

       2. associate-*r/ 88.4%

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

       3. associate-*l* 88.5%

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

       4. associate-/l/ 85.2%

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

       5. *-commutative 85.2%

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

       6. times-frac 88.5%

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

       7. *-inverses 88.5%

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

       8. *-lft-identity 88.5%

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

   12. Simplified 88.5%

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

   13. Taylor expanded in k around inf 88.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\frac{{\sin k}^{2}}{\cos k}}\right)\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3 \cdot {\left(\sqrt[3]{l\_m}\right)}^{2}}{t\_2} \cdot \frac{t\_m \cdot t\_3}{{\left(t\_m \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot t\_2\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k)))) (t_3 (/ (sqrt 2.0) k)))
   (*
    t_s
    (if (<= t_m 8e-108)
      (/
       2.0
       (*
        (/ (* t_m (pow k 2.0)) (cos k))
        (exp (+ (log (pow (sin k) 2.0)) (* -2.0 (log l_m))))))
      (*
       (/ (* t_3 (pow (cbrt l_m) 2.0)) t_2)
       (/ (* t_m t_3) (pow (* t_m (* (pow (cbrt l_m) -2.0) t_2)) 2.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (t_m <= 8e-108) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * exp((log(pow(sin(k), 2.0)) + (-2.0 * log(l_m)))));
	} else {
		tmp = ((t_3 * pow(cbrt(l_m), 2.0)) / t_2) * ((t_m * t_3) / pow((t_m * (pow(cbrt(l_m), -2.0) * t_2)), 2.0));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (t_m <= 8e-108) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * Math.exp((Math.log(Math.pow(Math.sin(k), 2.0)) + (-2.0 * Math.log(l_m)))));
	} else {
		tmp = ((t_3 * Math.pow(Math.cbrt(l_m), 2.0)) / t_2) * ((t_m * t_3) / Math.pow((t_m * (Math.pow(Math.cbrt(l_m), -2.0) * t_2)), 2.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (t_m <= 8e-108)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * exp(Float64(log((sin(k) ^ 2.0)) + Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(t_3 * (cbrt(l_m) ^ 2.0)) / t_2) * Float64(Float64(t_m * t_3) / (Float64(t_m * Float64((cbrt(l_m) ^ -2.0) * t_2)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-108], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000032e-108

   1. Initial program 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. pow2 78.0%

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

      2. add-log-exp 59.9%

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

      3. div-inv 59.8%

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

      4. exp-prod 64.1%

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

      5. pow2 64.1%

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

      6. pow-flip 64.1%

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

      7. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. pow-exp 59.8%

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

       2. rem-log-exp 78.0%

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

       3. pow-to-exp 35.6%

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

       4. pow-to-exp 17.3%

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

       5. prod-exp 18.9%

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

       6. rem-log-exp 18.9%

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

       7. pow-to-exp 44.3%

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

       8. rem-log-exp 40.9%

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

       9. pow-to-exp 76.6%

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

       10. log-pow 44.3%

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

   11. Applied egg-rr 44.3%

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

   if 8.00000000000000032e-108 < t

   1. Initial program 26.2%

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

   3. Simplified 26.2%

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

   6. Applied egg-rr 88.4%

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

      1. associate-/r/ 88.4%

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

      2. associate-/r* 88.4%

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

      3. associate-/r/ 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 87.5%

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

   10. Applied egg-rr 87.5%

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

       1. associate-/l* 88.4%

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

       2. associate-*r/ 88.4%

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

       3. associate-*l* 88.5%

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

       4. associate-/l/ 85.2%

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

       5. *-commutative 85.2%

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

       6. times-frac 88.5%

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

       7. *-inverses 88.5%

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

       8. *-lft-identity 88.5%

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

   12. Simplified 88.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{{\left(t\_m \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot t\_2\right)\right)}^{2}} \cdot \left(t\_3 \cdot \frac{{\left(\sqrt[3]{l\_m}\right)}^{2}}{t\_2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k)))) (t_3 (/ (sqrt 2.0) k)))
   (*
    t_s
    (if (<= t_m 7.5e-108)
      (/
       2.0
       (*
        (/ (* t_m (pow k 2.0)) (cos k))
        (exp (+ (log (pow (sin k) 2.0)) (* -2.0 (log l_m))))))
      (*
       (/ (* t_m t_3) (pow (* t_m (* (pow (cbrt l_m) -2.0) t_2)) 2.0))
       (* t_3 (/ (pow (cbrt l_m) 2.0) t_2)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (t_m <= 7.5e-108) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * exp((log(pow(sin(k), 2.0)) + (-2.0 * log(l_m)))));
	} else {
		tmp = ((t_m * t_3) / pow((t_m * (pow(cbrt(l_m), -2.0) * t_2)), 2.0)) * (t_3 * (pow(cbrt(l_m), 2.0) / t_2));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (t_m <= 7.5e-108) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * Math.exp((Math.log(Math.pow(Math.sin(k), 2.0)) + (-2.0 * Math.log(l_m)))));
	} else {
		tmp = ((t_m * t_3) / Math.pow((t_m * (Math.pow(Math.cbrt(l_m), -2.0) * t_2)), 2.0)) * (t_3 * (Math.pow(Math.cbrt(l_m), 2.0) / t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (t_m <= 7.5e-108)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * exp(Float64(log((sin(k) ^ 2.0)) + Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(t_m * t_3) / (Float64(t_m * Float64((cbrt(l_m) ^ -2.0) * t_2)) ^ 2.0)) * Float64(t_3 * Float64((cbrt(l_m) ^ 2.0) / t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-108], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.4999999999999993e-108

   1. Initial program 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in t around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. pow2 78.0%

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

      2. add-log-exp 59.9%

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

      3. div-inv 59.8%

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

      4. exp-prod 64.1%

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

      5. pow2 64.1%

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

      6. pow-flip 64.1%

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

      7. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. pow-exp 59.8%

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

       2. rem-log-exp 78.0%

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

       3. pow-to-exp 35.6%

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

       4. pow-to-exp 17.3%

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

       5. prod-exp 18.9%

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

       6. rem-log-exp 18.9%

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

       7. pow-to-exp 44.3%

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

       8. rem-log-exp 40.9%

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

       9. pow-to-exp 76.6%

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

       10. log-pow 44.3%

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

   11. Applied egg-rr 44.3%

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

   if 7.4999999999999993e-108 < t

   1. Initial program 26.2%

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

   3. Simplified 26.2%

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

   6. Applied egg-rr 88.4%

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

      1. associate-/r/ 88.4%

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

      2. associate-/r* 88.4%

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

      3. associate-/r/ 88.4%

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

   8. Simplified 88.4%

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

      1. associate-*r/ 87.5%

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

   10. Applied egg-rr 87.5%

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

       1. associate-/l* 88.4%

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

       2. associate-*r/ 88.4%

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

       3. associate-*l* 88.5%

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

       4. associate-/l/ 85.2%

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

       5. *-commutative 85.2%

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

       6. times-frac 88.5%

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

       7. *-inverses 88.5%

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

       8. *-lft-identity 88.5%

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

   12. Simplified 88.5%

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

       1. associate-*r/ 87.6%

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

       2. associate-/l* 89.5%

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

       3. associate-*r* 89.4%

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

   14. Applied egg-rr 89.4%

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

       1. associate-/l* 90.3%

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

       2. associate-*r/ 88.4%

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

       3. associate-*l* 88.5%

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

       4. associate-*r/ 88.4%

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

   16. Simplified 88.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_m \cdot t\_2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \frac{t\_2 \cdot {\left(\sqrt[3]{l\_m}\right)}^{2}}{\sqrt[3]{{k}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t\_m + 2 \cdot \log k}}{\cos k} \cdot \frac{{\sin k}^{2}}{{l\_m}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k)))
   (*
    t_s
    (if (<= l_m 2.3e-158)
      (*
       (/
        (* t_m t_2)
        (pow (* t_m (* (pow (cbrt l_m) -2.0) (cbrt (* (sin k) (tan k))))) 2.0))
       (/ (* t_2 (pow (cbrt l_m) 2.0)) (cbrt (pow k 2.0))))
      (/
       2.0
       (*
        (/ (exp (+ (log t_m) (* 2.0 (log k)))) (cos k))
        (/ (pow (sin k) 2.0) (pow l_m 2.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = sqrt(2.0) / k;
	double tmp;
	if (l_m <= 2.3e-158) {
		tmp = ((t_m * t_2) / pow((t_m * (pow(cbrt(l_m), -2.0) * cbrt((sin(k) * tan(k))))), 2.0)) * ((t_2 * pow(cbrt(l_m), 2.0)) / cbrt(pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((exp((log(t_m) + (2.0 * log(k)))) / cos(k)) * (pow(sin(k), 2.0) / pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.sqrt(2.0) / k;
	double tmp;
	if (l_m <= 2.3e-158) {
		tmp = ((t_m * t_2) / Math.pow((t_m * (Math.pow(Math.cbrt(l_m), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k))))), 2.0)) * ((t_2 * Math.pow(Math.cbrt(l_m), 2.0)) / Math.cbrt(Math.pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((Math.exp((Math.log(t_m) + (2.0 * Math.log(k)))) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / Math.pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (l_m <= 2.3e-158)
		tmp = Float64(Float64(Float64(t_m * t_2) / (Float64(t_m * Float64((cbrt(l_m) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))) ^ 2.0)) * Float64(Float64(t_2 * (cbrt(l_m) ^ 2.0)) / cbrt((k ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(exp(Float64(log(t_m) + Float64(2.0 * log(k)))) / cos(k)) * Float64((sin(k) ^ 2.0) / (l_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 2.3e-158], N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2999999999999999e-158

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   6. Applied egg-rr 86.3%

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

      1. associate-/r/ 86.3%

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

      2. associate-/r* 86.4%

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

      3. associate-/r/ 86.4%

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

   8. Simplified 86.4%

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

      1. associate-*r/ 85.9%

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

   10. Applied egg-rr 85.9%

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

       1. associate-/l* 86.4%

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

       2. associate-*r/ 86.4%

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

       3. associate-*l* 86.5%

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

       4. associate-/l/ 84.7%

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

       5. *-commutative 84.7%

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

       6. times-frac 86.5%

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

       7. *-inverses 86.5%

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

       8. *-lft-identity 86.5%

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

   12. Simplified 86.5%

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

   13. Taylor expanded in k around 0 78.0%

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

   if 2.2999999999999999e-158 < l

   1. Initial program 27.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. times-frac 83.6%

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

   7. Simplified 83.6%

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

      1. add-log-exp 37.8%

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

      2. *-commutative 37.8%

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

      3. exp-prod 28.1%

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

   9. Applied egg-rr 28.1%

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

       1. pow-exp 37.8%

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

       2. rem-log-exp 83.6%

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

       3. add-exp-log 34.0%

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

       4. pow-to-exp 15.6%

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

       5. prod-exp 16.4%

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

       6. rem-log-exp 15.6%

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

       7. pow-to-exp 33.9%

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

       8. log-pow 16.4%

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

   11. Applied egg-rr 16.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{{k}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t + 2 \cdot \log k}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{{l\_m}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left({k}^{2} \cdot \frac{t\_m}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t\_m + 2 \cdot \log k}}{\cos k} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (pow (sin k) 2.0) (pow l_m 2.0))))
   (*
    t_s
    (if (<= k 6e-161)
      (/ 2.0 (* (/ (* t_m (pow k 2.0)) (cos k)) 0.0))
      (if (<= k 2e+147)
        (/ 2.0 (* t_2 (* (pow k 2.0) (/ t_m (cos k)))))
        (/ 2.0 (* (/ (exp (+ (log t_m) (* 2.0 (log k)))) (cos k)) t_2)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(sin(k), 2.0) / pow(l_m, 2.0);
	double tmp;
	if (k <= 6e-161) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * 0.0);
	} else if (k <= 2e+147) {
		tmp = 2.0 / (t_2 * (pow(k, 2.0) * (t_m / cos(k))));
	} else {
		tmp = 2.0 / ((exp((log(t_m) + (2.0 * log(k)))) / cos(k)) * t_2);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sin(k) ** 2.0d0) / (l_m ** 2.0d0)
    if (k <= 6d-161) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * 0.0d0)
    else if (k <= 2d+147) then
        tmp = 2.0d0 / (t_2 * ((k ** 2.0d0) * (t_m / cos(k))))
    else
        tmp = 2.0d0 / ((exp((log(t_m) + (2.0d0 * log(k)))) / cos(k)) * t_2)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0) / Math.pow(l_m, 2.0);
	double tmp;
	if (k <= 6e-161) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * 0.0);
	} else if (k <= 2e+147) {
		tmp = 2.0 / (t_2 * (Math.pow(k, 2.0) * (t_m / Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.exp((Math.log(t_m) + (2.0 * Math.log(k)))) / Math.cos(k)) * t_2);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(math.sin(k), 2.0) / math.pow(l_m, 2.0)
	tmp = 0
	if k <= 6e-161:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * 0.0)
	elif k <= 2e+147:
		tmp = 2.0 / (t_2 * (math.pow(k, 2.0) * (t_m / math.cos(k))))
	else:
		tmp = 2.0 / ((math.exp((math.log(t_m) + (2.0 * math.log(k)))) / math.cos(k)) * t_2)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64((sin(k) ^ 2.0) / (l_m ^ 2.0))
	tmp = 0.0
	if (k <= 6e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * 0.0));
	elseif (k <= 2e+147)
		tmp = Float64(2.0 / Float64(t_2 * Float64((k ^ 2.0) * Float64(t_m / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(exp(Float64(log(t_m) + Float64(2.0 * log(k)))) / cos(k)) * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (sin(k) ^ 2.0) / (l_m ^ 2.0);
	tmp = 0.0;
	if (k <= 6e-161)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * 0.0);
	elseif (k <= 2e+147)
		tmp = 2.0 / (t_2 * ((k ^ 2.0) * (t_m / cos(k))));
	else
		tmp = 2.0 / ((exp((log(t_m) + (2.0 * log(k)))) / cos(k)) * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6e-161], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+147], N[(2.0 / N[(t$95$2 * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 5.99999999999999977e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 5.99999999999999977e-161 < k < 2e147

   1. Initial program 23.8%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in t around 0 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. times-frac 81.0%

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

   7. Simplified 81.0%

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

      1. add-log-exp 34.6%

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

      2. *-commutative 34.6%

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

      3. exp-prod 24.9%

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

   9. Applied egg-rr 24.9%

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

   10. Taylor expanded in t around 0 81.0%

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

       1. associate-/l* 81.1%

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

   12. Simplified 81.1%

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

   if 2e147 < 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%

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

   5. Taylor expanded in t around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. *-commutative 67.5%

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

      3. times-frac 67.5%

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

   7. Simplified 67.5%

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

      1. add-log-exp 67.5%

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

      2. *-commutative 67.5%

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

      3. exp-prod 38.7%

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

   9. Applied egg-rr 38.7%

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

       1. pow-exp 67.5%

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

       2. rem-log-exp 67.5%

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

       3. add-exp-log 43.4%

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

       4. pow-to-exp 43.4%

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

       5. prod-exp 47.4%

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

       6. rem-log-exp 43.4%

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

       7. pow-to-exp 43.4%

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

       8. log-pow 47.4%

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

   11. Applied egg-rr 47.4%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{{\ell}^{2}} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t + 2 \cdot \log k}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.2e+147)
    (/
     2.0
     (*
      (/ (* t_m (pow k 2.0)) (cos k))
      (exp (+ (log (pow (sin k) 2.0)) (* -2.0 (log l_m))))))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l_m) 2.0))
       (cbrt (* (sin k) (* (tan k) (pow (/ k t_m) 2.0)))))
      3.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.2e+147) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * exp((log(pow(sin(k), 2.0)) + (-2.0 * log(l_m)))));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l_m), 2.0)) * cbrt((sin(k) * (tan(k) * pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.2e+147) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * Math.exp((Math.log(Math.pow(Math.sin(k), 2.0)) + (-2.0 * Math.log(l_m)))));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l_m), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.2e+147)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * exp(Float64(log((sin(k) ^ 2.0)) + Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l_m) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e+147], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000001e147

   1. Initial program 28.0%

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

   3. Simplified 28.0%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 78.4%

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

   7. Simplified 78.4%

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

      1. pow2 78.4%

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

      2. add-log-exp 58.1%

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

      3. div-inv 58.1%

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

      4. exp-prod 61.1%

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

      5. pow2 61.1%

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

      6. pow-flip 61.1%

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

      7. metadata-eval 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. pow-exp 58.1%

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

       2. rem-log-exp 78.4%

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

       3. pow-to-exp 34.9%

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

       4. pow-to-exp 15.4%

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

       5. prod-exp 17.7%

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

       6. rem-log-exp 17.7%

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

       7. pow-to-exp 42.0%

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

       8. rem-log-exp 37.9%

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

       9. pow-to-exp 76.8%

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

       10. log-pow 42.0%

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

   11. Applied egg-rr 42.0%

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

   if 1.20000000000000001e147 < 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%

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

      1. add-cube-cbrt 38.1%

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

      2. pow3 38.1%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot e^{\log t\_2 + -2 \cdot \log l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t\_m + 2 \cdot \log k}}{\cos k} \cdot \frac{t\_2}{{l\_m}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 2e+147)
      (/
       2.0
       (*
        (/ (* t_m (pow k 2.0)) (cos k))
        (exp (+ (log t_2) (* -2.0 (log l_m))))))
      (/
       2.0
       (*
        (/ (exp (+ (log t_m) (* 2.0 (log k)))) (cos k))
        (/ t_2 (pow l_m 2.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 2e+147) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * exp((log(t_2) + (-2.0 * log(l_m)))));
	} else {
		tmp = 2.0 / ((exp((log(t_m) + (2.0 * log(k)))) / cos(k)) * (t_2 / pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (k <= 2d+147) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * exp((log(t_2) + ((-2.0d0) * log(l_m)))))
    else
        tmp = 2.0d0 / ((exp((log(t_m) + (2.0d0 * log(k)))) / cos(k)) * (t_2 / (l_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 2e+147) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * Math.exp((Math.log(t_2) + (-2.0 * Math.log(l_m)))));
	} else {
		tmp = 2.0 / ((Math.exp((Math.log(t_m) + (2.0 * Math.log(k)))) / Math.cos(k)) * (t_2 / Math.pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 2e+147:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * math.exp((math.log(t_2) + (-2.0 * math.log(l_m)))))
	else:
		tmp = 2.0 / ((math.exp((math.log(t_m) + (2.0 * math.log(k)))) / math.cos(k)) * (t_2 / math.pow(l_m, 2.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 2e+147)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * exp(Float64(log(t_2) + Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(exp(Float64(log(t_m) + Float64(2.0 * log(k)))) / cos(k)) * Float64(t_2 / (l_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 2e+147)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * exp((log(t_2) + (-2.0 * log(l_m)))));
	else
		tmp = 2.0 / ((exp((log(t_m) + (2.0 * log(k)))) / cos(k)) * (t_2 / (l_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2e+147], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[t$95$2], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e147

   1. Initial program 28.0%

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

   3. Simplified 28.0%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 78.4%

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

   7. Simplified 78.4%

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

      1. pow2 78.4%

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

      2. add-log-exp 58.1%

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

      3. div-inv 58.1%

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

      4. exp-prod 61.1%

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

      5. pow2 61.1%

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

      6. pow-flip 61.1%

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

      7. metadata-eval 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. pow-exp 58.1%

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

       2. rem-log-exp 78.4%

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

       3. pow-to-exp 34.9%

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

       4. pow-to-exp 15.4%

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

       5. prod-exp 17.7%

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

       6. rem-log-exp 17.7%

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

       7. pow-to-exp 42.0%

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

       8. rem-log-exp 37.9%

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

       9. pow-to-exp 76.8%

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

       10. log-pow 42.0%

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

   11. Applied egg-rr 42.0%

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

   if 2e147 < 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%

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

   5. Taylor expanded in t around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. *-commutative 67.5%

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

      3. times-frac 67.5%

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

   7. Simplified 67.5%

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

      1. add-log-exp 67.5%

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

      2. *-commutative 67.5%

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

      3. exp-prod 38.7%

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

   9. Applied egg-rr 38.7%

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

       1. pow-exp 67.5%

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

       2. rem-log-exp 67.5%

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

       3. add-exp-log 43.4%

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

       4. pow-to-exp 43.4%

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

       5. prod-exp 47.4%

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

       6. rem-log-exp 43.4%

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

       7. pow-to-exp 43.4%

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

       8. log-pow 47.4%

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

   11. Applied egg-rr 47.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot e^{\log \left({\sin k}^{2}\right) + -2 \cdot \log \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{e^{\log t + 2 \cdot \log k}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{{l\_m}^{2}} \cdot \left({k}^{2} \cdot \frac{t\_m}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 8e-161)
    (/ 2.0 (* (/ (* t_m (pow k 2.0)) (cos k)) 0.0))
    (/
     2.0
     (*
      (/ (pow (sin k) 2.0) (pow l_m 2.0))
      (* (pow k 2.0) (/ t_m (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 8e-161) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * 0.0);
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / pow(l_m, 2.0)) * (pow(k, 2.0) * (t_m / cos(k))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8d-161) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * 0.0d0)
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / (l_m ** 2.0d0)) * ((k ** 2.0d0) * (t_m / cos(k))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 8e-161) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * 0.0);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.pow(l_m, 2.0)) * (Math.pow(k, 2.0) * (t_m / Math.cos(k))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 8e-161:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * 0.0)
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.pow(l_m, 2.0)) * (math.pow(k, 2.0) * (t_m / math.cos(k))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 8e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * 0.0));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / (l_m ^ 2.0)) * Float64((k ^ 2.0) * Float64(t_m / cos(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 8e-161)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * 0.0);
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / (l_m ^ 2.0)) * ((k ^ 2.0) * (t_m / cos(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 8e-161], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000022e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 8.00000000000000022e-161 < k

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

      1. add-log-exp 42.4%

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

      2. *-commutative 42.4%

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

      3. exp-prod 28.1%

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

   9. Applied egg-rr 28.1%

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

   10. Taylor expanded in t around 0 77.8%

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

       1. associate-/l* 77.9%

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

   12. Simplified 77.9%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{{\ell}^{2}} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := t\_m \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t\_3}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{t\_2}{{l\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{t\_3 \cdot t\_2} \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)) (t_3 (* t_m (pow k 2.0))))
   (*
    t_s
    (if (<= k 3.3e-161)
      (/ 2.0 (* (/ t_3 (cos k)) 0.0))
      (if (<= k 2e-24)
        (/ 2.0 (* t_3 (/ t_2 (pow l_m 2.0))))
        (* (/ (* 2.0 (cos k)) (* t_3 t_2)) (* l_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(sin(k), 2.0);
	double t_3 = t_m * pow(k, 2.0);
	double tmp;
	if (k <= 3.3e-161) {
		tmp = 2.0 / ((t_3 / cos(k)) * 0.0);
	} else if (k <= 2e-24) {
		tmp = 2.0 / (t_3 * (t_2 / pow(l_m, 2.0)));
	} else {
		tmp = ((2.0 * cos(k)) / (t_3 * t_2)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    t_3 = t_m * (k ** 2.0d0)
    if (k <= 3.3d-161) then
        tmp = 2.0d0 / ((t_3 / cos(k)) * 0.0d0)
    else if (k <= 2d-24) then
        tmp = 2.0d0 / (t_3 * (t_2 / (l_m ** 2.0d0)))
    else
        tmp = ((2.0d0 * cos(k)) / (t_3 * t_2)) * (l_m * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = t_m * Math.pow(k, 2.0);
	double tmp;
	if (k <= 3.3e-161) {
		tmp = 2.0 / ((t_3 / Math.cos(k)) * 0.0);
	} else if (k <= 2e-24) {
		tmp = 2.0 / (t_3 * (t_2 / Math.pow(l_m, 2.0)));
	} else {
		tmp = ((2.0 * Math.cos(k)) / (t_3 * t_2)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(math.sin(k), 2.0)
	t_3 = t_m * math.pow(k, 2.0)
	tmp = 0
	if k <= 3.3e-161:
		tmp = 2.0 / ((t_3 / math.cos(k)) * 0.0)
	elif k <= 2e-24:
		tmp = 2.0 / (t_3 * (t_2 / math.pow(l_m, 2.0)))
	else:
		tmp = ((2.0 * math.cos(k)) / (t_3 * t_2)) * (l_m * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(t_m * (k ^ 2.0))
	tmp = 0.0
	if (k <= 3.3e-161)
		tmp = Float64(2.0 / Float64(Float64(t_3 / cos(k)) * 0.0));
	elseif (k <= 2e-24)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_2 / (l_m ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(t_3 * t_2)) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0;
	t_3 = t_m * (k ^ 2.0);
	tmp = 0.0;
	if (k <= 3.3e-161)
		tmp = 2.0 / ((t_3 / cos(k)) * 0.0);
	elseif (k <= 2e-24)
		tmp = 2.0 / (t_3 * (t_2 / (l_m ^ 2.0)));
	else
		tmp = ((2.0 * cos(k)) / (t_3 * t_2)) * (l_m * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.3e-161], N[(2.0 / N[(N[(t$95$3 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e-24], N[(2.0 / N[(t$95$3 * N[(t$95$2 / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.2999999999999998e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 3.2999999999999998e-161 < k < 1.99999999999999985e-24

   1. Initial program 25.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in t around 0 65.9%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in k around 0 78.0%

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

   if 1.99999999999999985e-24 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. associate-*r/ 77.6%

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

      2. associate-*r* 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(t \cdot {k}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := t\_m \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t\_3}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 10^{-24}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{t\_2}{{l\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t\_m \cdot t\_2\right)}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)) (t_3 (* t_m (pow k 2.0))))
   (*
    t_s
    (if (<= k 5.5e-161)
      (/ 2.0 (* (/ t_3 (cos k)) 0.0))
      (if (<= k 1e-24)
        (/ 2.0 (* t_3 (/ t_2 (pow l_m 2.0))))
        (* (* l_m l_m) (/ 2.0 (/ (* (* k k) (* t_m t_2)) (cos k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(sin(k), 2.0);
	double t_3 = t_m * pow(k, 2.0);
	double tmp;
	if (k <= 5.5e-161) {
		tmp = 2.0 / ((t_3 / cos(k)) * 0.0);
	} else if (k <= 1e-24) {
		tmp = 2.0 / (t_3 * (t_2 / pow(l_m, 2.0)));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * t_2)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    t_3 = t_m * (k ** 2.0d0)
    if (k <= 5.5d-161) then
        tmp = 2.0d0 / ((t_3 / cos(k)) * 0.0d0)
    else if (k <= 1d-24) then
        tmp = 2.0d0 / (t_3 * (t_2 / (l_m ** 2.0d0)))
    else
        tmp = (l_m * l_m) * (2.0d0 / (((k * k) * (t_m * t_2)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = t_m * Math.pow(k, 2.0);
	double tmp;
	if (k <= 5.5e-161) {
		tmp = 2.0 / ((t_3 / Math.cos(k)) * 0.0);
	} else if (k <= 1e-24) {
		tmp = 2.0 / (t_3 * (t_2 / Math.pow(l_m, 2.0)));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * t_2)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(math.sin(k), 2.0)
	t_3 = t_m * math.pow(k, 2.0)
	tmp = 0
	if k <= 5.5e-161:
		tmp = 2.0 / ((t_3 / math.cos(k)) * 0.0)
	elif k <= 1e-24:
		tmp = 2.0 / (t_3 * (t_2 / math.pow(l_m, 2.0)))
	else:
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * t_2)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(t_m * (k ^ 2.0))
	tmp = 0.0
	if (k <= 5.5e-161)
		tmp = Float64(2.0 / Float64(Float64(t_3 / cos(k)) * 0.0));
	elseif (k <= 1e-24)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_2 / (l_m ^ 2.0))));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m * t_2)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = sin(k) ^ 2.0;
	t_3 = t_m * (k ^ 2.0);
	tmp = 0.0;
	if (k <= 5.5e-161)
		tmp = 2.0 / ((t_3 / cos(k)) * 0.0);
	elseif (k <= 1e-24)
		tmp = 2.0 / (t_3 * (t_2 / (l_m ^ 2.0)));
	else
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * t_2)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5.5e-161], N[(2.0 / N[(N[(t$95$3 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1e-24], N[(2.0 / N[(t$95$3 * N[(t$95$2 / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.5e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 5.5e-161 < k < 9.99999999999999924e-25

   1. Initial program 25.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in t around 0 65.9%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in k around 0 78.0%

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

   if 9.99999999999999924e-25 < k

   1. Initial program 28.3%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. unpow2 77.7%

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

   7. Applied egg-rr 77.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 10^{-24}:\\ \;\;\;\;\frac{2}{\left(t \cdot {k}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{{l\_m}^{2}} \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-161)
    (/ 2.0 (* (/ (* t_m (pow k 2.0)) (cos k)) 0.0))
    (/
     2.0
     (* (/ (pow (sin k) 2.0) (pow l_m 2.0)) (/ (* t_m (* k k)) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-161) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * 0.0);
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / pow(l_m, 2.0)) * ((t_m * (k * k)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.8d-161) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * 0.0d0)
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / (l_m ** 2.0d0)) * ((t_m * (k * k)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-161) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * 0.0);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.pow(l_m, 2.0)) * ((t_m * (k * k)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.8e-161:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * 0.0)
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.pow(l_m, 2.0)) * ((t_m * (k * k)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.8e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * 0.0));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / (l_m ^ 2.0)) * Float64(Float64(t_m * Float64(k * k)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.8e-161)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * 0.0);
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / (l_m ^ 2.0)) * ((t_m * (k * k)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.8e-161], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000009e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 1.80000000000000009e-161 < k

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

      1. unpow2 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{l\_m}^{2}}{{\sin k}^{2}} \cdot \frac{2 \cdot \cos k}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-160)
    (/ 2.0 (* (/ (* t_m (pow k 2.0)) (cos k)) 0.0))
    (*
     (/ (pow l_m 2.0) (pow (sin k) 2.0))
     (/ (* 2.0 (cos k)) (* t_m (* k k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1e-160) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * 0.0);
	} else {
		tmp = (pow(l_m, 2.0) / pow(sin(k), 2.0)) * ((2.0 * cos(k)) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-160) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * 0.0d0)
    else
        tmp = ((l_m ** 2.0d0) / (sin(k) ** 2.0d0)) * ((2.0d0 * cos(k)) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1e-160) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * 0.0);
	} else {
		tmp = (Math.pow(l_m, 2.0) / Math.pow(Math.sin(k), 2.0)) * ((2.0 * Math.cos(k)) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1e-160:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * 0.0)
	else:
		tmp = (math.pow(l_m, 2.0) / math.pow(math.sin(k), 2.0)) * ((2.0 * math.cos(k)) / (t_m * (k * k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1e-160)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * 0.0));
	else
		tmp = Float64(Float64((l_m ^ 2.0) / (sin(k) ^ 2.0)) * Float64(Float64(2.0 * cos(k)) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1e-160)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * 0.0);
	else
		tmp = ((l_m ^ 2.0) / (sin(k) ^ 2.0)) * ((2.0 * cos(k)) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1e-160], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999999e-161

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. add-log-exp 62.3%

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

      3. div-inv 62.2%

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

      4. exp-prod 67.8%

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

      5. pow2 67.8%

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

      6. pow-flip 67.8%

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

      7. metadata-eval 67.8%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in k around 0 47.6%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 9.9999999999999999e-161 < k

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

   6. Applied egg-rr 82.4%

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

      1. associate-/r/ 82.4%

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

      2. associate-/r* 82.4%

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

      3. associate-/r/ 83.5%

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

   8. Simplified 83.5%

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

   9. Taylor expanded in k around inf 73.3%

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

       1. associate-*r* 73.3%

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

       2. *-commutative 73.3%

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

       3. *-commutative 73.3%

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

       4. times-frac 77.5%

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

       5. *-commutative 77.5%

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

       6. unpow2 77.5%

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

       7. rem-square-sqrt 77.8%

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

   11. Simplified 77.8%

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

       1. unpow2 77.8%

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

   13. Applied egg-rr 77.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{2 \cdot \cos k}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.26e-129)
    (/ 2.0 (* (/ (* t_m (pow k 2.0)) (cos k)) 0.0))
    (*
     (* l_m l_m)
     (/ 2.0 (/ (* (* k k) (* t_m (pow (sin k) 2.0))) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.26e-129) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * 0.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * pow(sin(k), 2.0))) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.26d-129) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / cos(k)) * 0.0d0)
    else
        tmp = (l_m * l_m) * (2.0d0 / (((k * k) * (t_m * (sin(k) ** 2.0d0))) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.26e-129) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * 0.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * Math.pow(Math.sin(k), 2.0))) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.26e-129:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * 0.0)
	else:
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * math.pow(math.sin(k), 2.0))) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.26e-129)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * 0.0));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m * (sin(k) ^ 2.0))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.26e-129)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * 0.0);
	else
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (sin(k) ^ 2.0))) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.26e-129], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2599999999999999e-129

   1. Initial program 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. pow2 78.0%

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

      2. add-log-exp 63.7%

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

      3. div-inv 63.6%

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

      4. exp-prod 68.9%

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

      5. pow2 68.9%

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

      6. pow-flip 68.9%

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

      7. metadata-eval 68.9%

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

   9. Applied egg-rr 68.9%

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

   10. Taylor expanded in k around 0 49.7%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 1.2599999999999999e-129 < k

   1. Initial program 26.5%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in t around 0 71.7%

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

      1. unpow2 76.5%

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

   7. Applied egg-rr 71.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-5}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{t\_2 \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* t_m (pow k 2.0))))
   (*
    t_s
    (if (<= k 1.35e-129)
      (/ 2.0 (* (/ t_2 (cos k)) 0.0))
      (if (<= k 2.05e-5)
        (* (* l_m l_m) (/ 2.0 (/ (* t_2 (* k k)) (cos k))))
        (*
         (* l_m l_m)
         (/
          2.0
          (/
           (* (* k k) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))
           (cos k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = t_m * pow(k, 2.0);
	double tmp;
	if (k <= 1.35e-129) {
		tmp = 2.0 / ((t_2 / cos(k)) * 0.0);
	} else if (k <= 2.05e-5) {
		tmp = (l_m * l_m) * (2.0 / ((t_2 * (k * k)) / cos(k)));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k ** 2.0d0)
    if (k <= 1.35d-129) then
        tmp = 2.0d0 / ((t_2 / cos(k)) * 0.0d0)
    else if (k <= 2.05d-5) then
        tmp = (l_m * l_m) * (2.0d0 / ((t_2 * (k * k)) / cos(k)))
    else
        tmp = (l_m * l_m) * (2.0d0 / (((k * k) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = t_m * Math.pow(k, 2.0);
	double tmp;
	if (k <= 1.35e-129) {
		tmp = 2.0 / ((t_2 / Math.cos(k)) * 0.0);
	} else if (k <= 2.05e-5) {
		tmp = (l_m * l_m) * (2.0 / ((t_2 * (k * k)) / Math.cos(k)));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = t_m * math.pow(k, 2.0)
	tmp = 0
	if k <= 1.35e-129:
		tmp = 2.0 / ((t_2 / math.cos(k)) * 0.0)
	elif k <= 2.05e-5:
		tmp = (l_m * l_m) * (2.0 / ((t_2 * (k * k)) / math.cos(k)))
	else:
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(t_m * (k ^ 2.0))
	tmp = 0.0
	if (k <= 1.35e-129)
		tmp = Float64(2.0 / Float64(Float64(t_2 / cos(k)) * 0.0));
	elseif (k <= 2.05e-5)
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(t_2 * Float64(k * k)) / cos(k))));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = t_m * (k ^ 2.0);
	tmp = 0.0;
	if (k <= 1.35e-129)
		tmp = 2.0 / ((t_2 / cos(k)) * 0.0);
	elseif (k <= 2.05e-5)
		tmp = (l_m * l_m) * (2.0 / ((t_2 * (k * k)) / cos(k)));
	else
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := Block[{t$95$2 = N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.35e-129], N[(2.0 / N[(N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e-5], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(t$95$2 * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.35e-129

   1. Initial program 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. pow2 78.0%

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

      2. add-log-exp 63.7%

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

      3. div-inv 63.6%

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

      4. exp-prod 68.9%

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

      5. pow2 68.9%

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

      6. pow-flip 68.9%

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

      7. metadata-eval 68.9%

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

   9. Applied egg-rr 68.9%

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

   10. Taylor expanded in k around 0 49.7%

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \log \color{blue}{1}} \]

   if 1.35e-129 < k < 2.05000000000000002e-5

   1. Initial program 21.8%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in t around 0 64.5%

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

      1. unpow2 78.3%

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

   7. Applied egg-rr 64.5%

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

   8. Taylor expanded in k around 0 64.4%

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

   if 2.05000000000000002e-5 < k

   1. Initial program 29.0%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 75.5%

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

      1. unpow2 75.5%

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

      2. sin-mult 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. div-sub 75.1%

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

      2. +-inverses 75.1%

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

      3. cos-0 75.1%

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

      4. metadata-eval 75.1%

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

      5. count-2 75.1%

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

   9. Simplified 75.1%

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

       1. unpow2 75.6%

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

   11. Applied egg-rr 75.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot 0}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-5}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-5}:\\ \;\;\;\;\left(l\_m \cdot \frac{l\_m}{t\_m}\right) \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.05e-5)
    (* (* l_m (/ l_m t_m)) (/ 2.0 (pow k 4.0)))
    (*
     (* l_m l_m)
     (/
      2.0
      (/ (* (* k k) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0)))) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.05e-5) {
		tmp = (l_m * (l_m / t_m)) * (2.0 / pow(k, 4.0));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.05d-5) then
        tmp = (l_m * (l_m / t_m)) * (2.0d0 / (k ** 4.0d0))
    else
        tmp = (l_m * l_m) * (2.0d0 / (((k * k) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.05e-5) {
		tmp = (l_m * (l_m / t_m)) * (2.0 / Math.pow(k, 4.0));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.05e-5:
		tmp = (l_m * (l_m / t_m)) * (2.0 / math.pow(k, 4.0))
	else:
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.05e-5)
		tmp = Float64(Float64(l_m * Float64(l_m / t_m)) * Float64(2.0 / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.05e-5)
		tmp = (l_m * (l_m / t_m)) * (2.0 / (k ^ 4.0));
	else
		tmp = (l_m * l_m) * (2.0 / (((k * k) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2.05e-5], N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.05000000000000002e-5

   1. Initial program 28.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in k around 0 64.2%

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

      1. associate-*r/ 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

      4. times-frac 63.4%

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

   7. Simplified 63.4%

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

      1. pow2 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. associate-/l* 68.1%

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

   11. Applied egg-rr 68.1%

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

   if 2.05000000000000002e-5 < k

   1. Initial program 29.0%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 75.5%

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

      1. unpow2 75.5%

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

      2. sin-mult 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. div-sub 75.1%

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

      2. +-inverses 75.1%

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

      3. cos-0 75.1%

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

      4. metadata-eval 75.1%

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

      5. count-2 75.1%

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

   9. Simplified 75.1%

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

       1. unpow2 75.6%

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

   11. Applied egg-rr 75.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-5}:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\left(l\_m \cdot \frac{l\_m}{t\_m}\right) \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 0.0)
    (* (* l_m (/ l_m t_m)) (/ 2.0 (pow k 4.0)))
    (* (* l_m l_m) (/ 2.0 (/ (* (* t_m (pow k 2.0)) (* k k)) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = (l_m * (l_m / t_m)) * (2.0 / pow(k, 4.0));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((t_m * pow(k, 2.0)) * (k * k)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l_m * l_m) <= 0.0d0) then
        tmp = (l_m * (l_m / t_m)) * (2.0d0 / (k ** 4.0d0))
    else
        tmp = (l_m * l_m) * (2.0d0 / (((t_m * (k ** 2.0d0)) * (k * k)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = (l_m * (l_m / t_m)) * (2.0 / Math.pow(k, 4.0));
	} else {
		tmp = (l_m * l_m) * (2.0 / (((t_m * Math.pow(k, 2.0)) * (k * k)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (l_m * l_m) <= 0.0:
		tmp = (l_m * (l_m / t_m)) * (2.0 / math.pow(k, 4.0))
	else:
		tmp = (l_m * l_m) * (2.0 / (((t_m * math.pow(k, 2.0)) * (k * k)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(Float64(l_m * Float64(l_m / t_m)) * Float64(2.0 / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * Float64(k * k)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((l_m * l_m) <= 0.0)
		tmp = (l_m * (l_m / t_m)) * (2.0 / (k ^ 4.0));
	else
		tmp = (l_m * l_m) * (2.0 / (((t_m * (k ^ 2.0)) * (k * k)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 25.9%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in k around 0 54.8%

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

      1. associate-*r/ 54.8%

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

      2. *-commutative 54.8%

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

      3. *-commutative 54.8%

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

      4. times-frac 54.9%

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

   7. Simplified 54.9%

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

      1. pow2 54.9%

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

   9. Applied egg-rr 54.9%

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

       1. associate-/l* 66.6%

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

   11. Applied egg-rr 66.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 29.9%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in t around 0 80.7%

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

      1. unpow2 84.6%

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

   7. Applied egg-rr 80.7%

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

   8. Taylor expanded in k around 0 65.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $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$95$m_, k_] := N[(t$95$s * N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 37.6%

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

5. Taylor expanded in k around 0 60.6%

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

   1. associate-*r/ 60.6%

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

   2. *-commutative 60.6%

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

   3. *-commutative 60.6%

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

   4. times-frac 59.9%

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

7. Simplified 59.9%

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

   1. pow2 59.9%

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

9. Applied egg-rr 59.9%

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

    1. associate-/l* 63.8%

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

11. Applied egg-rr 63.8%

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

Reproduce

herbie shell --seed 2024137 
(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))))