Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 81.6%

Time: 25.9s

Alternatives: 18

Speedup: 2.0×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k_m \cdot \tan k_m\\ t_3 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k_m}{t_m}\right)\right)\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;{\left(\frac{t_4}{t_m \cdot {\left(\sqrt[3]{k_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(\sqrt{t_2} \cdot t_3\right)\right)}^{2}}\\ \mathbf{elif}\;k_m \leq 1800000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{t_4}{t_m \cdot \sqrt[3]{{k_m}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 2.15 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot {\sin k_m}^{2}}\\ \mathbf{elif}\;k_m \leq 1.25 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{2}{t_2 \cdot {t_m}^{3}}}{\frac{t_3}{\ell}} \cdot \frac{\ell}{t_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (sin k_m) (tan k_m)))
        (t_3 (hypot 1.0 (hypot 1.0 (/ k_m t_m))))
        (t_4 (pow (cbrt l) 2.0)))
   (*
    t_s
    (if (<= k_m 3.2e-87)
      (pow (/ t_4 (* t_m (pow (cbrt k_m) 2.0))) 3.0)
      (if (<= k_m 6.5e-51)
        (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* (sqrt t_2) t_3)) 2.0))
        (if (<= k_m 1800000000.0)
          (pow (cbrt (pow (/ t_4 (* t_m (cbrt (pow k_m 2.0)))) 3.0)) 3.0)
          (if (<= k_m 2.15e+150)
            (*
             2.0
             (/
              (* (pow l 2.0) (cos k_m))
              (* (* t_m (pow k_m 2.0)) (pow (sin k_m) 2.0))))
            (if (<= k_m 1.25e+215)
              (* (/ (/ 2.0 (* t_2 (pow t_m 3.0))) (/ t_3 l)) (/ l t_3))
              (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = hypot(1.0, hypot(1.0, (k_m / t_m)));
	double t_4 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 3.2e-87) {
		tmp = pow((t_4 / (t_m * pow(cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 6.5e-51) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (sqrt(t_2) * t_3)), 2.0);
	} else if (k_m <= 1800000000.0) {
		tmp = pow(cbrt(pow((t_4 / (t_m * cbrt(pow(k_m, 2.0)))), 3.0)), 3.0);
	} else if (k_m <= 2.15e+150) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * pow(sin(k_m), 2.0)));
	} else if (k_m <= 1.25e+215) {
		tmp = ((2.0 / (t_2 * pow(t_m, 3.0))) / (t_3 / l)) * (l / t_3);
	} else {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)));
	double t_4 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 3.2e-87) {
		tmp = Math.pow((t_4 / (t_m * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 6.5e-51) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.sqrt(t_2) * t_3)), 2.0);
	} else if (k_m <= 1800000000.0) {
		tmp = Math.pow(Math.cbrt(Math.pow((t_4 / (t_m * Math.cbrt(Math.pow(k_m, 2.0)))), 3.0)), 3.0);
	} else if (k_m <= 2.15e+150) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0)));
	} else if (k_m <= 1.25e+215) {
		tmp = ((2.0 / (t_2 * Math.pow(t_m, 3.0))) / (t_3 / l)) * (l / t_3);
	} else {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = hypot(1.0, hypot(1.0, Float64(k_m / t_m)))
	t_4 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 3.2e-87)
		tmp = Float64(t_4 / Float64(t_m * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	elseif (k_m <= 6.5e-51)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(sqrt(t_2) * t_3)) ^ 2.0));
	elseif (k_m <= 1800000000.0)
		tmp = cbrt((Float64(t_4 / Float64(t_m * cbrt((k_m ^ 2.0)))) ^ 3.0)) ^ 3.0;
	elseif (k_m <= 2.15e+150)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	elseif (k_m <= 1.25e+215)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_2 * (t_m ^ 3.0))) / Float64(t_3 / l)) * Float64(l / t_3));
	else
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 3.2e-87], N[Power[N[(t$95$4 / N[(t$95$m * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 6.5e-51], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1800000000.0], N[Power[N[Power[N[Power[N[(t$95$4 / N[(t$95$m * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 2.15e+150], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.25e+215], N[(N[(N[(2.0 / N[(t$95$2 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 6 regimes

2. if k < 3.19999999999999979e-87

   1. Initial program 58.9%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in k around 0 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 53.2%

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

   6. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.1%

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

      2. pow2 53.1%

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

      3. cbrt-div 53.1%

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

      4. pow2 53.1%

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

      5. cbrt-prod 53.1%

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

      6. pow2 53.1%

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

      7. cbrt-prod 53.0%

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

      8. unpow3 53.0%

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

      9. add-cbrt-cube 53.1%

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

      10. unpow2 53.1%

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

      11. cbrt-prod 53.1%

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

      12. pow2 53.1%

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

      13. cbrt-div 53.1%

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

      14. pow2 53.1%

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

      15. cbrt-prod 58.4%

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

      16. pow2 58.4%

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

   8. Applied egg-rr 81.8%

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

      1. pow-plus 81.9%

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

      2. metadata-eval 81.9%

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

   10. Simplified 81.9%

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

   if 3.19999999999999979e-87 < k < 6.5000000000000003e-51

   1. Initial program 71.8%

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

      1. add-sqr-sqrt 29.2%

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

      2. pow2 29.2%

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

      3. sqrt-div 29.2%

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

      4. sqrt-pow1 29.2%

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

      5. metadata-eval 29.2%

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

      6. sqrt-prod 15.1%

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

      7. add-sqr-sqrt 29.2%

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

   3. Applied egg-rr 29.2%

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

      1. expm1-log1p-u 29.2%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      2. expm1-udef 15.9%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} - 1}} \]

   5. Applied egg-rr 27.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} - 1}} \]
   6. Step-by-step derivation

      1. expm1-def 41.2%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      2. expm1-log1p 42.5%

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

      3. associate-*l* 42.5%

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

   7. Simplified 42.5%

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

   if 6.5000000000000003e-51 < k < 1.8e9

   1. Initial program 64.6%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

   6. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 73.4%

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

      2. pow2 73.4%

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

      3. cbrt-div 73.5%

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

      4. pow2 73.5%

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

      5. cbrt-prod 73.6%

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

      6. pow2 73.6%

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

      7. cbrt-prod 73.4%

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

      8. unpow3 73.4%

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

      9. add-cbrt-cube 73.4%

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

      10. unpow2 73.4%

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

      11. cbrt-prod 73.5%

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

      12. pow2 73.5%

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

      13. cbrt-div 73.6%

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

      14. pow2 73.6%

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

      15. cbrt-prod 82.1%

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

      16. pow2 82.1%

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

   8. Applied egg-rr 82.0%

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

      1. pow-plus 82.0%

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

      2. metadata-eval 82.0%

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

   10. Simplified 82.0%

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

       1. add-cbrt-cube 82.0%

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

       2. pow3 82.0%

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

       3. unpow2 82.0%

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

       4. cbrt-prod 81.6%

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

       5. unpow2 81.6%

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

   12. Applied egg-rr 81.6%

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

   if 1.8e9 < k < 2.14999999999999999e150

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in t around 0 94.9%

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

      1. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

   if 2.14999999999999999e150 < k < 1.25e215

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

      1. associate-*r* 53.2%

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

      2. add-sqr-sqrt 53.2%

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

      3. times-frac 53.4%

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

   5. Applied egg-rr 60.3%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*r* 73.0%

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

   7. Simplified 73.0%

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

   if 1.25e215 < k

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in t around 0 75.7%

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

   5. Taylor expanded in k around 0 75.7%

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

      1. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. expm1-log1p-u 75.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 75.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 75.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 75.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 75.7%

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

   11. Simplified 75.7%

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

       1. add-sqr-sqrt 75.7%

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

       2. pow2 75.7%

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

       3. sqrt-prod 75.7%

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

       4. unpow2 75.7%

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

       5. sqrt-prod 33.6%

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

       6. add-sqr-sqrt 76.0%

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

       7. sqrt-pow1 76.0%

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

       8. metadata-eval 76.0%

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

   13. Applied egg-rr 76.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1800000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \end{array} \]

Alternative 2: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k_m}{t_m}\right)\right)\\ t_3 := \sin k_m \cdot \tan k_m\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 4.1 \cdot 10^{-167}:\\ \;\;\;\;{\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k_m} \cdot \sqrt{t_m}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 24000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(\sqrt{t_3} \cdot \frac{{t_m}^{1.5}}{\ell}\right) \cdot t_2}\right)}^{2}\\ \mathbf{elif}\;k_m \leq 7.4 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot {\sin k_m}^{2}}\\ \mathbf{elif}\;k_m \leq 1.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{2}{t_3 \cdot {t_m}^{3}}}{\frac{t_2}{\ell}} \cdot \frac{\ell}{t_2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k_m t_m))))
        (t_3 (* (sin k_m) (tan k_m))))
   (*
    t_s
    (if (<= k_m 4.1e-167)
      (pow (pow (/ (cbrt l) (* (cbrt k_m) (sqrt t_m))) 2.0) 3.0)
      (if (<= k_m 24000000000.0)
        (pow (/ (sqrt 2.0) (* (* (sqrt t_3) (/ (pow t_m 1.5) l)) t_2)) 2.0)
        (if (<= k_m 7.4e+149)
          (*
           2.0
           (/
            (* (pow l 2.0) (cos k_m))
            (* (* t_m (pow k_m 2.0)) (pow (sin k_m) 2.0))))
          (if (<= k_m 1.75e+215)
            (* (/ (/ 2.0 (* t_3 (pow t_m 3.0))) (/ t_2 l)) (/ l t_2))
            (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m)))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = hypot(1.0, hypot(1.0, (k_m / t_m)));
	double t_3 = sin(k_m) * tan(k_m);
	double tmp;
	if (k_m <= 4.1e-167) {
		tmp = pow(pow((cbrt(l) / (cbrt(k_m) * sqrt(t_m))), 2.0), 3.0);
	} else if (k_m <= 24000000000.0) {
		tmp = pow((sqrt(2.0) / ((sqrt(t_3) * (pow(t_m, 1.5) / l)) * t_2)), 2.0);
	} else if (k_m <= 7.4e+149) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * pow(sin(k_m), 2.0)));
	} else if (k_m <= 1.75e+215) {
		tmp = ((2.0 / (t_3 * pow(t_m, 3.0))) / (t_2 / l)) * (l / t_2);
	} else {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)));
	double t_3 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if (k_m <= 4.1e-167) {
		tmp = Math.pow(Math.pow((Math.cbrt(l) / (Math.cbrt(k_m) * Math.sqrt(t_m))), 2.0), 3.0);
	} else if (k_m <= 24000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.sqrt(t_3) * (Math.pow(t_m, 1.5) / l)) * t_2)), 2.0);
	} else if (k_m <= 7.4e+149) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0)));
	} else if (k_m <= 1.75e+215) {
		tmp = ((2.0 / (t_3 * Math.pow(t_m, 3.0))) / (t_2 / l)) * (l / t_2);
	} else {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = hypot(1.0, hypot(1.0, Float64(k_m / t_m)))
	t_3 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (k_m <= 4.1e-167)
		tmp = (Float64(cbrt(l) / Float64(cbrt(k_m) * sqrt(t_m))) ^ 2.0) ^ 3.0;
	elseif (k_m <= 24000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(sqrt(t_3) * Float64((t_m ^ 1.5) / l)) * t_2)) ^ 2.0;
	elseif (k_m <= 7.4e+149)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	elseif (k_m <= 1.75e+215)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_3 * (t_m ^ 3.0))) / Float64(t_2 / l)) * Float64(l / t_2));
	else
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4.1e-167], N[Power[N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 24000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 7.4e+149], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.75e+215], N[(N[(N[(2.0 / N[(t$95$3 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if k < 4.10000000000000018e-167

   1. Initial program 58.7%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in k around 0 51.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 51.9%

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

   6. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 51.9%

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

      2. pow2 51.9%

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

      3. cbrt-div 51.9%

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

      4. pow2 51.9%

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

      5. cbrt-prod 51.9%

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

      6. pow2 51.9%

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

      7. cbrt-prod 51.8%

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

      8. unpow3 51.8%

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

      9. add-cbrt-cube 51.8%

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

      10. unpow2 51.8%

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

      11. cbrt-prod 51.8%

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

      12. pow2 51.8%

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

      13. cbrt-div 51.8%

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

      14. pow2 51.8%

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

      15. cbrt-prod 56.4%

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

      16. pow2 56.4%

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

   8. Applied egg-rr 80.7%

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

      1. pow-plus 80.7%

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

      2. metadata-eval 80.7%

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

   10. Simplified 80.7%

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

       1. add-exp-log 40.9%

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

   12. Applied egg-rr 40.9%

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

       1. rem-exp-log 80.7%

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

       2. associate-/r* 80.7%

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

       3. unpow2 80.7%

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

       4. cbrt-prod 67.5%

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

       5. unpow2 67.5%

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

       6. associate-/r* 67.6%

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

       7. expm1-log1p-u 47.8%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)\right)\right)}}^{3} \]

       8. expm1-udef 44.9%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)} - 1\right)}}^{3} \]

   14. Applied egg-rr 35.0%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right)}}^{3} \]
   15. Step-by-step derivation

       1. expm1-def 38.3%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)\right)\right)}}^{3} \]

       2. expm1-log1p 38.4%

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

   16. Simplified 38.4%

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

   if 4.10000000000000018e-167 < k < 2.4e10

   1. Initial program 64.4%

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

   3. Simplified 64.3%

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

   5. Applied egg-rr 46.8%

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

   if 2.4e10 < k < 7.39999999999999957e149

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in t around 0 94.9%

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

      1. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

   if 7.39999999999999957e149 < k < 1.74999999999999988e215

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

      1. associate-*r* 53.2%

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

      2. add-sqr-sqrt 53.2%

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

      3. times-frac 53.4%

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

   5. Applied egg-rr 60.3%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*r* 73.0%

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

   7. Simplified 73.0%

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

   if 1.74999999999999988e215 < k

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in t around 0 75.7%

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

   5. Taylor expanded in k around 0 75.7%

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

      1. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. expm1-log1p-u 75.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 75.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 75.7%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 75.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 75.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 75.7%

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

   11. Simplified 75.7%

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

       1. add-sqr-sqrt 75.7%

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

       2. pow2 75.7%

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

       3. sqrt-prod 75.7%

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

       4. unpow2 75.7%

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

       5. sqrt-prod 33.6%

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

       6. add-sqr-sqrt 76.0%

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

       7. sqrt-pow1 76.0%

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

       8. metadata-eval 76.0%

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

   13. Applied egg-rr 76.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-167}:\\ \;\;\;\;{\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)}^{3}\\ \mathbf{elif}\;k \leq 24000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \end{array} \]

Alternative 3: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;{\left(\frac{t_2}{t_m \cdot {\left(\sqrt[3]{k_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt{\sin k_m \cdot \tan k_m} \cdot \frac{{t_m}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k_m}{t_m}\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k_m \leq 1800000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{t_2}{t_m \cdot \sqrt[3]{{k_m}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot {\sin k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0)))
   (*
    t_s
    (if (<= k_m 5.2e-88)
      (pow (/ t_2 (* t_m (pow (cbrt k_m) 2.0))) 3.0)
      (if (<= k_m 1.65e-52)
        (/
         2.0
         (pow
          (*
           (* (sqrt (* (sin k_m) (tan k_m))) (/ (pow t_m 1.5) l))
           (hypot 1.0 (hypot 1.0 (/ k_m t_m))))
          2.0))
        (if (<= k_m 1800000000.0)
          (pow (cbrt (pow (/ t_2 (* t_m (cbrt (pow k_m 2.0)))) 3.0)) 3.0)
          (*
           2.0
           (/
            (* (pow l 2.0) (cos k_m))
            (* (* t_m (pow k_m 2.0)) (pow (sin k_m) 2.0))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 5.2e-88) {
		tmp = pow((t_2 / (t_m * pow(cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.65e-52) {
		tmp = 2.0 / pow(((sqrt((sin(k_m) * tan(k_m))) * (pow(t_m, 1.5) / l)) * hypot(1.0, hypot(1.0, (k_m / t_m)))), 2.0);
	} else if (k_m <= 1800000000.0) {
		tmp = pow(cbrt(pow((t_2 / (t_m * cbrt(pow(k_m, 2.0)))), 3.0)), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 5.2e-88) {
		tmp = Math.pow((t_2 / (t_m * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.65e-52) {
		tmp = 2.0 / Math.pow(((Math.sqrt((Math.sin(k_m) * Math.tan(k_m))) * (Math.pow(t_m, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)))), 2.0);
	} else if (k_m <= 1800000000.0) {
		tmp = Math.pow(Math.cbrt(Math.pow((t_2 / (t_m * Math.cbrt(Math.pow(k_m, 2.0)))), 3.0)), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 5.2e-88)
		tmp = Float64(t_2 / Float64(t_m * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	elseif (k_m <= 1.65e-52)
		tmp = Float64(2.0 / (Float64(Float64(sqrt(Float64(sin(k_m) * tan(k_m))) * Float64((t_m ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) ^ 2.0));
	elseif (k_m <= 1800000000.0)
		tmp = cbrt((Float64(t_2 / Float64(t_m * cbrt((k_m ^ 2.0)))) ^ 3.0)) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 5.2e-88], N[Power[N[(t$95$2 / N[(t$95$m * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 1.65e-52], N[(2.0 / N[Power[N[(N[(N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1800000000.0], N[Power[N[Power[N[Power[N[(t$95$2 / N[(t$95$m * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 5.20000000000000027e-88

   1. Initial program 58.9%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in k around 0 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 53.2%

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

   6. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.1%

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

      2. pow2 53.1%

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

      3. cbrt-div 53.1%

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

      4. pow2 53.1%

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

      5. cbrt-prod 53.1%

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

      6. pow2 53.1%

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

      7. cbrt-prod 53.0%

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

      8. unpow3 53.0%

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

      9. add-cbrt-cube 53.1%

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

      10. unpow2 53.1%

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

      11. cbrt-prod 53.1%

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

      12. pow2 53.1%

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

      13. cbrt-div 53.1%

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

      14. pow2 53.1%

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

      15. cbrt-prod 58.4%

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

      16. pow2 58.4%

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

   8. Applied egg-rr 81.8%

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

      1. pow-plus 81.9%

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

      2. metadata-eval 81.9%

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

   10. Simplified 81.9%

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

   if 5.20000000000000027e-88 < k < 1.64999999999999998e-52

   1. Initial program 71.8%

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

      1. add-sqr-sqrt 29.0%

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

      2. pow2 29.0%

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

   3. Applied egg-rr 42.5%

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

   if 1.64999999999999998e-52 < k < 1.8e9

   1. Initial program 64.6%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

   6. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 73.4%

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

      2. pow2 73.4%

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

      3. cbrt-div 73.5%

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

      4. pow2 73.5%

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

      5. cbrt-prod 73.6%

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

      6. pow2 73.6%

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

      7. cbrt-prod 73.4%

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

      8. unpow3 73.4%

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

      9. add-cbrt-cube 73.4%

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

      10. unpow2 73.4%

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

      11. cbrt-prod 73.5%

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

      12. pow2 73.5%

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

      13. cbrt-div 73.6%

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

      14. pow2 73.6%

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

      15. cbrt-prod 82.1%

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

      16. pow2 82.1%

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

   8. Applied egg-rr 82.0%

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

      1. pow-plus 82.0%

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

      2. metadata-eval 82.0%

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

   10. Simplified 82.0%

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

       1. add-cbrt-cube 82.0%

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

       2. pow3 82.0%

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

       3. unpow2 82.0%

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

       4. cbrt-prod 81.6%

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

       5. unpow2 81.6%

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

   12. Applied egg-rr 81.6%

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

   if 1.8e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1800000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 4: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 6 \cdot 10^{-88}:\\ \;\;\;\;{\left(\frac{t_2}{t_m \cdot {\left(\sqrt[3]{k_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 7.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k_m \cdot \tan k_m} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k_m}{t_m}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k_m \leq 2900000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{t_2}{t_m \cdot \sqrt[3]{{k_m}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot {\sin k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0)))
   (*
    t_s
    (if (<= k_m 6e-88)
      (pow (/ t_2 (* t_m (pow (cbrt k_m) 2.0))) 3.0)
      (if (<= k_m 7.4e-50)
        (/
         2.0
         (pow
          (*
           (/ (pow t_m 1.5) l)
           (*
            (sqrt (* (sin k_m) (tan k_m)))
            (hypot 1.0 (hypot 1.0 (/ k_m t_m)))))
          2.0))
        (if (<= k_m 2900000000.0)
          (pow (cbrt (pow (/ t_2 (* t_m (cbrt (pow k_m 2.0)))) 3.0)) 3.0)
          (*
           2.0
           (/
            (* (pow l 2.0) (cos k_m))
            (* (* t_m (pow k_m 2.0)) (pow (sin k_m) 2.0))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 6e-88) {
		tmp = pow((t_2 / (t_m * pow(cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 7.4e-50) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (sqrt((sin(k_m) * tan(k_m))) * hypot(1.0, hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 2900000000.0) {
		tmp = pow(cbrt(pow((t_2 / (t_m * cbrt(pow(k_m, 2.0)))), 3.0)), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 6e-88) {
		tmp = Math.pow((t_2 / (t_m * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 7.4e-50) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.sqrt((Math.sin(k_m) * Math.tan(k_m))) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 2900000000.0) {
		tmp = Math.pow(Math.cbrt(Math.pow((t_2 / (t_m * Math.cbrt(Math.pow(k_m, 2.0)))), 3.0)), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 6e-88)
		tmp = Float64(t_2 / Float64(t_m * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	elseif (k_m <= 7.4e-50)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(sqrt(Float64(sin(k_m) * tan(k_m))) * hypot(1.0, hypot(1.0, Float64(k_m / t_m))))) ^ 2.0));
	elseif (k_m <= 2900000000.0)
		tmp = cbrt((Float64(t_2 / Float64(t_m * cbrt((k_m ^ 2.0)))) ^ 3.0)) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6e-88], N[Power[N[(t$95$2 / N[(t$95$m * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 7.4e-50], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2900000000.0], N[Power[N[Power[N[Power[N[(t$95$2 / N[(t$95$m * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 5.9999999999999999e-88

   1. Initial program 58.9%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in k around 0 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 53.2%

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

   6. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.1%

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

      2. pow2 53.1%

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

      3. cbrt-div 53.1%

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

      4. pow2 53.1%

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

      5. cbrt-prod 53.1%

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

      6. pow2 53.1%

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

      7. cbrt-prod 53.0%

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

      8. unpow3 53.0%

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

      9. add-cbrt-cube 53.1%

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

      10. unpow2 53.1%

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

      11. cbrt-prod 53.1%

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

      12. pow2 53.1%

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

      13. cbrt-div 53.1%

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

      14. pow2 53.1%

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

      15. cbrt-prod 58.4%

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

      16. pow2 58.4%

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

   8. Applied egg-rr 81.8%

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

      1. pow-plus 81.9%

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

      2. metadata-eval 81.9%

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

   10. Simplified 81.9%

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

   if 5.9999999999999999e-88 < k < 7.4000000000000002e-50

   1. Initial program 71.8%

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

      1. add-sqr-sqrt 29.2%

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

      2. pow2 29.2%

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

      3. sqrt-div 29.2%

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

      4. sqrt-pow1 29.2%

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

      5. metadata-eval 29.2%

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

      6. sqrt-prod 15.1%

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

      7. add-sqr-sqrt 29.2%

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

   3. Applied egg-rr 29.2%

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

      1. expm1-log1p-u 29.2%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      2. expm1-udef 15.9%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} - 1}} \]

   5. Applied egg-rr 27.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)} - 1}} \]
   6. Step-by-step derivation

      1. expm1-def 41.2%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)\right)}} \]

      2. expm1-log1p 42.5%

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

      3. associate-*l* 42.5%

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

   7. Simplified 42.5%

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

   if 7.4000000000000002e-50 < k < 2.9e9

   1. Initial program 64.6%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

   6. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 73.4%

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

      2. pow2 73.4%

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

      3. cbrt-div 73.5%

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

      4. pow2 73.5%

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

      5. cbrt-prod 73.6%

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

      6. pow2 73.6%

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

      7. cbrt-prod 73.4%

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

      8. unpow3 73.4%

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

      9. add-cbrt-cube 73.4%

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

      10. unpow2 73.4%

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

      11. cbrt-prod 73.5%

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

      12. pow2 73.5%

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

      13. cbrt-div 73.6%

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

      14. pow2 73.6%

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

      15. cbrt-prod 82.1%

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

      16. pow2 82.1%

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

   8. Applied egg-rr 82.0%

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

      1. pow-plus 82.0%

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

      2. metadata-eval 82.0%

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

   10. Simplified 82.0%

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

       1. add-cbrt-cube 82.0%

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

       2. pow3 82.0%

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

       3. unpow2 82.0%

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

       4. cbrt-prod 81.6%

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

       5. unpow2 81.6%

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

   12. Applied egg-rr 81.6%

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

   if 2.9e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-88}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2900000000:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 5: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 12500000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t_m \cdot {\left(\sqrt[3]{k_m}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot {\sin k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 12500000000.0)
    (pow (/ (pow (cbrt l) 2.0) (* t_m (pow (cbrt k_m) 2.0))) 3.0)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k_m))
      (* (* t_m (pow k_m 2.0)) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 12500000000.0) {
		tmp = pow((pow(cbrt(l), 2.0) / (t_m * pow(cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 12500000000.0) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 12500000000.0)
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 12500000000.0], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e10

   1. Initial program 59.7%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 55.1%

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

   6. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 55.0%

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

      2. pow2 55.0%

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

      3. cbrt-div 55.0%

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

      4. pow2 55.0%

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

      5. cbrt-prod 55.0%

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

      6. pow2 55.0%

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

      7. cbrt-prod 54.9%

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

      8. unpow3 54.9%

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

      9. add-cbrt-cube 54.9%

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

      10. unpow2 54.9%

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

      11. cbrt-prod 54.9%

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

      12. pow2 54.9%

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

      13. cbrt-div 55.0%

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

      14. pow2 55.0%

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

      15. cbrt-prod 60.8%

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

      16. pow2 60.8%

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

   8. Applied egg-rr 82.4%

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

      1. pow-plus 82.4%

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

      2. metadata-eval 82.4%

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

   10. Simplified 82.4%

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

   if 1.25e10 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 12500000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 6: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1800000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t_m \cdot {\left(\sqrt[3]{k_m}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k_m \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1800000000.0)
    (pow (/ (pow (cbrt l) 2.0) (* t_m (pow (cbrt k_m) 2.0))) 3.0)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k_m))
      (* (* t_m (pow k_m 2.0)) (+ 0.5 (* -0.5 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1800000000.0) {
		tmp = pow((pow(cbrt(l), 2.0) / (t_m * pow(cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * (0.5 + (-0.5 * cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1800000000.0) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1800000000.0)
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1800000000.0], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e9

   1. Initial program 59.7%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 55.1%

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

   6. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 55.0%

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

      2. pow2 55.0%

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

      3. cbrt-div 55.0%

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

      4. pow2 55.0%

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

      5. cbrt-prod 55.0%

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

      6. pow2 55.0%

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

      7. cbrt-prod 54.9%

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

      8. unpow3 54.9%

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

      9. add-cbrt-cube 54.9%

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

      10. unpow2 54.9%

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

      11. cbrt-prod 54.9%

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

      12. pow2 54.9%

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

      13. cbrt-div 55.0%

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

      14. pow2 55.0%

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

      15. cbrt-prod 60.8%

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

      16. pow2 60.8%

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

   8. Applied egg-rr 82.4%

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

      1. pow-plus 82.4%

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

      2. metadata-eval 82.4%

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

   10. Simplified 82.4%

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

   if 1.8e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. unpow2 75.8%

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

      2. sin-mult 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-sub 75.8%

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

      2. +-inverses 75.8%

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

      3. cos-0 75.8%

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

      4. metadata-eval 75.8%

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

      5. count-2 75.8%

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

      6. *-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in k around inf 75.8%

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

       1. associate-*r* 75.8%

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

       2. cancel-sign-sub-inv 75.8%

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

       3. metadata-eval 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1800000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}\\ \end{array} \]

Alternative 7: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1800000000:\\ \;\;\;\;{\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k_m} \cdot \sqrt{t_m}}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k_m \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1800000000.0)
    (pow (pow (/ (cbrt l) (* (cbrt k_m) (sqrt t_m))) 2.0) 3.0)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k_m))
      (* (* t_m (pow k_m 2.0)) (+ 0.5 (* -0.5 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1800000000.0) {
		tmp = pow(pow((cbrt(l) / (cbrt(k_m) * sqrt(t_m))), 2.0), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * (0.5 + (-0.5 * cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1800000000.0) {
		tmp = Math.pow(Math.pow((Math.cbrt(l) / (Math.cbrt(k_m) * Math.sqrt(t_m))), 2.0), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1800000000.0)
		tmp = (Float64(cbrt(l) / Float64(cbrt(k_m) * sqrt(t_m))) ^ 2.0) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1800000000.0], N[Power[N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e9

   1. Initial program 59.7%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 55.1%

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

   6. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 55.0%

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

      2. pow2 55.0%

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

      3. cbrt-div 55.0%

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

      4. pow2 55.0%

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

      5. cbrt-prod 55.0%

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

      6. pow2 55.0%

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

      7. cbrt-prod 54.9%

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

      8. unpow3 54.9%

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

      9. add-cbrt-cube 54.9%

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

      10. unpow2 54.9%

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

      11. cbrt-prod 54.9%

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

      12. pow2 54.9%

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

      13. cbrt-div 55.0%

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

      14. pow2 55.0%

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

      15. cbrt-prod 60.8%

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

      16. pow2 60.8%

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

   8. Applied egg-rr 82.4%

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

      1. pow-plus 82.4%

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

      2. metadata-eval 82.4%

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

   10. Simplified 82.4%

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

       1. add-exp-log 42.1%

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

   12. Applied egg-rr 42.1%

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

       1. rem-exp-log 82.4%

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

       2. associate-/r* 82.4%

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

       3. unpow2 82.4%

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

       4. cbrt-prod 71.5%

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

       5. unpow2 71.5%

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

       6. associate-/r* 71.5%

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

       7. expm1-log1p-u 52.5%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)\right)\right)}}^{3} \]

       8. expm1-udef 48.3%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)} - 1\right)}}^{3} \]

   14. Applied egg-rr 35.5%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right)}}^{3} \]
   15. Step-by-step derivation

       1. expm1-def 39.1%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)\right)\right)}}^{3} \]

       2. expm1-log1p 39.3%

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

   16. Simplified 39.3%

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

   if 1.8e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. unpow2 75.8%

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

      2. sin-mult 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-sub 75.8%

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

      2. +-inverses 75.8%

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

      3. cos-0 75.8%

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

      4. metadata-eval 75.8%

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

      5. count-2 75.8%

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

      6. *-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in k around inf 75.8%

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

       1. associate-*r* 75.8%

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

       2. cancel-sign-sub-inv 75.8%

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

       3. metadata-eval 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1800000000:\\ \;\;\;\;{\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt{t}}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}\\ \end{array} \]

Alternative 8: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t_m \cdot {k_m}^{0.6666666666666666}}\right)}^{3}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.5 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k_m \leq 8.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{{\left(k_m \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k_m}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k_m \leq 1800000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k_m}^{2}} \cdot \frac{\cos k_m}{t_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k_m \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2
         (pow
          (/ (pow (cbrt l) 2.0) (* t_m (pow k_m 0.6666666666666666)))
          3.0)))
   (*
    t_s
    (if (<= k_m 1.5e-185)
      t_2
      (if (<= k_m 8.2e-53)
        (/
         2.0
         (*
          (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)
          (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0)))))
        (if (<= k_m 1800000000.0)
          t_2
          (*
           2.0
           (*
            (/ (pow l 2.0) (pow k_m 2.0))
            (/ (cos k_m) (* t_m (- 0.5 (* 0.5 (cos (* k_m 2.0))))))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow((pow(cbrt(l), 2.0) / (t_m * pow(k_m, 0.6666666666666666))), 3.0);
	double tmp;
	if (k_m <= 1.5e-185) {
		tmp = t_2;
	} else if (k_m <= 8.2e-53) {
		tmp = 2.0 / (pow((k_m * (pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + pow((k_m / t_m), 2.0))));
	} else if (k_m <= 1800000000.0) {
		tmp = t_2;
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * (0.5 - (0.5 * cos((k_m * 2.0)))))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(k_m, 0.6666666666666666))), 3.0);
	double tmp;
	if (k_m <= 1.5e-185) {
		tmp = t_2;
	} else if (k_m <= 8.2e-53) {
		tmp = 2.0 / (Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0))));
	} else if (k_m <= 1800000000.0) {
		tmp = t_2;
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * (0.5 - (0.5 * Math.cos((k_m * 2.0)))))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (k_m ^ 0.6666666666666666))) ^ 3.0
	tmp = 0.0
	if (k_m <= 1.5e-185)
		tmp = t_2;
	elseif (k_m <= 8.2e-53)
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))));
	elseif (k_m <= 1800000000.0)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0))))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.5e-185], t$95$2, If[LessEqual[k$95$m, 8.2e-53], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1800000000.0], t$95$2, N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000015e-185 or 8.2000000000000001e-53 < k < 1.8e9

   1. Initial program 59.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 53.8%

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

   6. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.7%

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

      2. pow2 53.7%

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

      3. cbrt-div 53.7%

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

      4. pow2 53.7%

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

      5. cbrt-prod 53.7%

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

      6. pow2 53.7%

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

      7. cbrt-prod 53.6%

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

      8. unpow3 53.6%

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

      9. add-cbrt-cube 53.7%

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

      10. unpow2 53.7%

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

      11. cbrt-prod 53.7%

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

      12. pow2 53.7%

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

      13. cbrt-div 53.7%

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

      14. pow2 53.7%

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

      15. cbrt-prod 58.6%

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

      16. pow2 58.6%

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

   8. Applied egg-rr 80.3%

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

      1. pow-plus 80.3%

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

      2. metadata-eval 80.3%

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

   10. Simplified 80.3%

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

       1. expm1-log1p-u 57.4%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\right)\right)}}^{3} \]

       2. expm1-udef 52.0%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)} - 1\right)}}^{3} \]

       3. unpow2 52.0%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}}\right)} - 1\right)}^{3} \]

       4. cbrt-prod 46.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\sqrt[3]{k \cdot k}}}\right)} - 1\right)}^{3} \]

       5. unpow2 46.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\color{blue}{{k}^{2}}}}\right)} - 1\right)}^{3} \]

   12. Applied egg-rr 46.4%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)} - 1\right)}}^{3} \]
   13. Step-by-step derivation

       1. expm1-def 49.2%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)\right)\right)}}^{3} \]

       2. expm1-log1p 69.3%

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

   14. Simplified 69.3%

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

       1. pow1/3 68.7%

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

       2. pow-pow 21.1%

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

       3. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{\color{blue}{0.6666666666666666}}}\right)}^{3} \]

   16. Applied egg-rr 21.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{{k}^{0.6666666666666666}}}\right)}^{3} \]

   if 1.50000000000000015e-185 < k < 8.2000000000000001e-53

   1. Initial program 63.7%

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

      1. associate-*r* 59.8%

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

      2. add-sqr-sqrt 33.7%

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

      3. pow2 33.7%

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

      4. *-commutative 33.7%

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

      5. sqrt-prod 33.7%

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

      6. sqrt-div 37.2%

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

      7. sqrt-pow1 44.7%

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

      8. metadata-eval 44.7%

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

      9. sqrt-prod 29.5%

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

      10. add-sqr-sqrt 55.5%

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

   3. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in k around 0 59.0%

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

   if 1.8e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. unpow2 75.8%

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

      2. sin-mult 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-sub 75.8%

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

      2. +-inverses 75.8%

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

      3. cos-0 75.8%

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

      4. metadata-eval 75.8%

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

      5. count-2 75.8%

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

      6. *-commutative 75.8%

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

   8. Simplified 75.8%

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

      1. times-frac 70.3%

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

      2. div-inv 70.3%

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

      3. *-commutative 70.3%

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

      4. metadata-eval 70.3%

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

   10. Applied egg-rr 70.3%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-185}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{0.6666666666666666}}\right)}^{3}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1800000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{0.6666666666666666}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \end{array} \]

Alternative 9: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t_m \cdot {k_m}^{0.6666666666666666}}\right)}^{3}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k_m \leq 1.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{{\left(k_m \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k_m}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k_m \leq 1800000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{\left(t_m \cdot {k_m}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k_m \cdot 2\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2
         (pow
          (/ (pow (cbrt l) 2.0) (* t_m (pow k_m 0.6666666666666666)))
          3.0)))
   (*
    t_s
    (if (<= k_m 1.3e-185)
      t_2
      (if (<= k_m 1.4e-50)
        (/
         2.0
         (*
          (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)
          (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0)))))
        (if (<= k_m 1800000000.0)
          t_2
          (*
           2.0
           (/
            (* (pow l 2.0) (cos k_m))
            (*
             (* t_m (pow k_m 2.0))
             (+ 0.5 (* -0.5 (cos (* k_m 2.0)))))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow((pow(cbrt(l), 2.0) / (t_m * pow(k_m, 0.6666666666666666))), 3.0);
	double tmp;
	if (k_m <= 1.3e-185) {
		tmp = t_2;
	} else if (k_m <= 1.4e-50) {
		tmp = 2.0 / (pow((k_m * (pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + pow((k_m / t_m), 2.0))));
	} else if (k_m <= 1800000000.0) {
		tmp = t_2;
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / ((t_m * pow(k_m, 2.0)) * (0.5 + (-0.5 * cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(k_m, 0.6666666666666666))), 3.0);
	double tmp;
	if (k_m <= 1.3e-185) {
		tmp = t_2;
	} else if (k_m <= 1.4e-50) {
		tmp = 2.0 / (Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0))));
	} else if (k_m <= 1800000000.0) {
		tmp = t_2;
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / ((t_m * Math.pow(k_m, 2.0)) * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (k_m ^ 0.6666666666666666))) ^ 3.0
	tmp = 0.0
	if (k_m <= 1.3e-185)
		tmp = t_2;
	elseif (k_m <= 1.4e-50)
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))));
	elseif (k_m <= 1800000000.0)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.3e-185], t$95$2, If[LessEqual[k$95$m, 1.4e-50], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1800000000.0], t$95$2, N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.29999999999999992e-185 or 1.3999999999999999e-50 < k < 1.8e9

   1. Initial program 59.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 53.8%

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

   6. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.7%

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

      2. pow2 53.7%

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

      3. cbrt-div 53.7%

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

      4. pow2 53.7%

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

      5. cbrt-prod 53.7%

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

      6. pow2 53.7%

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

      7. cbrt-prod 53.6%

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

      8. unpow3 53.6%

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

      9. add-cbrt-cube 53.7%

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

      10. unpow2 53.7%

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

      11. cbrt-prod 53.7%

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

      12. pow2 53.7%

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

      13. cbrt-div 53.7%

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

      14. pow2 53.7%

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

      15. cbrt-prod 58.6%

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

      16. pow2 58.6%

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

   8. Applied egg-rr 80.3%

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

      1. pow-plus 80.3%

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

      2. metadata-eval 80.3%

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

   10. Simplified 80.3%

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

       1. expm1-log1p-u 57.4%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\right)\right)}}^{3} \]

       2. expm1-udef 52.0%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)} - 1\right)}}^{3} \]

       3. unpow2 52.0%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}}\right)} - 1\right)}^{3} \]

       4. cbrt-prod 46.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\sqrt[3]{k \cdot k}}}\right)} - 1\right)}^{3} \]

       5. unpow2 46.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\color{blue}{{k}^{2}}}}\right)} - 1\right)}^{3} \]

   12. Applied egg-rr 46.4%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)} - 1\right)}}^{3} \]
   13. Step-by-step derivation

       1. expm1-def 49.2%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)\right)\right)}}^{3} \]

       2. expm1-log1p 69.3%

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

   14. Simplified 69.3%

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

       1. pow1/3 68.7%

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

       2. pow-pow 21.1%

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

       3. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{\color{blue}{0.6666666666666666}}}\right)}^{3} \]

   16. Applied egg-rr 21.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{{k}^{0.6666666666666666}}}\right)}^{3} \]

   if 1.29999999999999992e-185 < k < 1.3999999999999999e-50

   1. Initial program 63.7%

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

      1. associate-*r* 59.8%

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

      2. add-sqr-sqrt 33.7%

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

      3. pow2 33.7%

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

      4. *-commutative 33.7%

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

      5. sqrt-prod 33.7%

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

      6. sqrt-div 37.2%

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

      7. sqrt-pow1 44.7%

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

      8. metadata-eval 44.7%

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

      9. sqrt-prod 29.5%

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

      10. add-sqr-sqrt 55.5%

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

   3. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in k around 0 59.0%

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

   if 1.8e9 < k

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 75.8%

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

      1. unpow2 75.8%

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

      2. sin-mult 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-sub 75.8%

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

      2. +-inverses 75.8%

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

      3. cos-0 75.8%

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

      4. metadata-eval 75.8%

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

      5. count-2 75.8%

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

      6. *-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in k around inf 75.8%

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

       1. associate-*r* 75.8%

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

       2. cancel-sign-sub-inv 75.8%

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

       3. metadata-eval 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{0.6666666666666666}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 1800000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{0.6666666666666666}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5 \cdot 10^{-113}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{elif}\;t_m \leq 1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{{\left(k_m \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k_m}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t_m \cdot {k_m}^{0.6666666666666666}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-113)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (if (<= t_m 1.05e+155)
      (/
       2.0
       (*
        (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)
        (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0)))))
      (pow (/ (pow (cbrt l) 2.0) (* t_m (pow k_m 0.6666666666666666))) 3.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5e-113) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else if (t_m <= 1.05e+155) {
		tmp = 2.0 / (pow((k_m * (pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + pow((k_m / t_m), 2.0))));
	} else {
		tmp = pow((pow(cbrt(l), 2.0) / (t_m * pow(k_m, 0.6666666666666666))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5e-113) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else if (t_m <= 1.05e+155) {
		tmp = 2.0 / (Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0))));
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(k_m, 0.6666666666666666))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5e-113)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	elseif (t_m <= 1.05e+155)
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))));
	else
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (k_m ^ 0.6666666666666666))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-113], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+155], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.9999999999999997e-113

   1. Initial program 52.3%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in t around 0 64.6%

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

   5. Taylor expanded in k around 0 52.9%

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

      1. associate-/r* 53.5%

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

   7. Simplified 53.5%

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

      1. expm1-log1p-u 53.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 53.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 53.2%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 53.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 53.5%

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

   11. Simplified 53.5%

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

       1. add-sqr-sqrt 53.5%

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

       2. pow2 53.5%

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

       3. sqrt-prod 53.5%

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

       4. unpow2 53.5%

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

       5. sqrt-prod 31.8%

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

       6. add-sqr-sqrt 59.5%

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

       7. sqrt-pow1 60.2%

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

       8. metadata-eval 60.2%

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

   13. Applied egg-rr 60.2%

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

   if 4.9999999999999997e-113 < t < 1.05e155

   1. Initial program 59.9%

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

      1. associate-*r* 59.5%

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

      2. add-sqr-sqrt 38.1%

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

      3. pow2 38.1%

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

      4. *-commutative 38.1%

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

      5. sqrt-prod 38.1%

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

      6. sqrt-div 41.2%

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

      7. sqrt-pow1 55.9%

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

      8. metadata-eval 55.9%

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

      9. sqrt-prod 34.4%

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

      10. add-sqr-sqrt 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

   5. Simplified 62.2%

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

   6. Taylor expanded in k around 0 82.8%

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

   if 1.05e155 < t

   1. Initial program 74.5%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in k around 0 57.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 57.3%

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

   6. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 57.3%

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

      2. pow2 57.3%

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

      3. cbrt-div 57.3%

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

      4. pow2 57.3%

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

      5. cbrt-prod 57.3%

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

      6. pow2 57.3%

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

      7. cbrt-prod 57.3%

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

      8. unpow3 57.3%

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

      9. add-cbrt-cube 57.3%

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

      10. unpow2 57.3%

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

      11. cbrt-prod 57.3%

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

      12. pow2 57.3%

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

      13. cbrt-div 57.3%

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

      14. pow2 57.3%

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

      15. cbrt-prod 61.6%

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

      16. pow2 61.6%

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

   8. Applied egg-rr 94.5%

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

      1. pow-plus 94.5%

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

      2. metadata-eval 94.5%

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

   10. Simplified 94.5%

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

       1. expm1-log1p-u 94.1%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\right)\right)}}^{3} \]

       2. expm1-udef 86.7%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)} - 1\right)}}^{3} \]

       3. unpow2 86.7%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}}\right)} - 1\right)}^{3} \]

       4. cbrt-prod 66.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{\sqrt[3]{k \cdot k}}}\right)} - 1\right)}^{3} \]

       5. unpow2 66.4%

          \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\color{blue}{{k}^{2}}}}\right)} - 1\right)}^{3} \]

   12. Applied egg-rr 66.4%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)} - 1\right)}}^{3} \]
   13. Step-by-step derivation

       1. expm1-def 66.3%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{{k}^{2}}}\right)\right)\right)}}^{3} \]

       2. expm1-log1p 66.3%

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

   14. Simplified 66.3%

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

       1. pow1/3 66.3%

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

       2. pow-pow 36.6%

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

       3. metadata-eval 36.6%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{\color{blue}{0.6666666666666666}}}\right)}^{3} \]

   16. Applied egg-rr 36.6%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \color{blue}{{k}^{0.6666666666666666}}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-113}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {k}^{0.6666666666666666}}\right)}^{3}\\ \end{array} \]

Alternative 11: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.2 \cdot 10^{-113}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k_m \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k_m}{t_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-113)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (/
     2.0
     (*
      (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)
      (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.2e-113) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 / (pow((k_m * (pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + pow((k_m / t_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.2d-113) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 / (((k_m * ((t_m ** 1.5d0) / l)) ** 2.0d0) * (1.0d0 + (1.0d0 + ((k_m / t_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.2e-113) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 / (Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3.2e-113:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 / (math.pow((k_m * (math.pow(t_m, 1.5) / l)), 2.0) * (1.0 + (1.0 + math.pow((k_m / t_m), 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3.2e-113)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3.2e-113)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 / (((k_m * ((t_m ^ 1.5) / l)) ^ 2.0) * (1.0 + (1.0 + ((k_m / t_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.2e-113], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.2000000000000002e-113

   1. Initial program 52.3%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in t around 0 64.6%

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

   5. Taylor expanded in k around 0 52.9%

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

      1. associate-/r* 53.5%

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

   7. Simplified 53.5%

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

      1. expm1-log1p-u 53.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 53.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 53.2%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 53.2%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 53.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 53.5%

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

   11. Simplified 53.5%

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

       1. add-sqr-sqrt 53.5%

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

       2. pow2 53.5%

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

       3. sqrt-prod 53.5%

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

       4. unpow2 53.5%

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

       5. sqrt-prod 31.8%

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

       6. add-sqr-sqrt 59.5%

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

       7. sqrt-pow1 60.2%

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

       8. metadata-eval 60.2%

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

   13. Applied egg-rr 60.2%

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

   if 3.2000000000000002e-113 < t

   1. Initial program 64.0%

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

      1. associate-*r* 58.9%

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

      2. add-sqr-sqrt 37.7%

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

      3. pow2 37.7%

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

      4. *-commutative 37.7%

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

      5. sqrt-prod 37.6%

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

      6. sqrt-div 39.9%

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

      7. sqrt-pow1 50.4%

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

      8. metadata-eval 50.4%

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

      9. sqrt-prod 31.5%

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

      10. add-sqr-sqrt 55.0%

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

   3. Applied egg-rr 55.0%

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

      1. *-commutative 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in k around 0 81.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-113}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 12: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{k_m}^{2} \cdot {t_m}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{\frac{t_m}{{k_m}^{-4}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 3.8e-150)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (if (<= l 3.9e+150)
      (/ 2.0 (* 2.0 (/ (* (pow k_m 2.0) (pow t_m 3.0)) (pow l 2.0))))
      (* 2.0 (pow (/ l (sqrt (/ t_m (pow k_m -4.0)))) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 3.8e-150) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else if (l <= 3.9e+150) {
		tmp = 2.0 / (2.0 * ((pow(k_m, 2.0) * pow(t_m, 3.0)) / pow(l, 2.0)));
	} else {
		tmp = 2.0 * pow((l / sqrt((t_m / pow(k_m, -4.0)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 3.8d-150) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else if (l <= 3.9d+150) then
        tmp = 2.0d0 / (2.0d0 * (((k_m ** 2.0d0) * (t_m ** 3.0d0)) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 * ((l / sqrt((t_m / (k_m ** (-4.0d0))))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 3.8e-150) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else if (l <= 3.9e+150) {
		tmp = 2.0 / (2.0 * ((Math.pow(k_m, 2.0) * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 * Math.pow((l / Math.sqrt((t_m / Math.pow(k_m, -4.0)))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 3.8e-150:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	elif l <= 3.9e+150:
		tmp = 2.0 / (2.0 * ((math.pow(k_m, 2.0) * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 * math.pow((l / math.sqrt((t_m / math.pow(k_m, -4.0)))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 3.8e-150)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	elseif (l <= 3.9e+150)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64((k_m ^ 2.0) * (t_m ^ 3.0)) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 * (Float64(l / sqrt(Float64(t_m / (k_m ^ -4.0)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 3.8e-150)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	elseif (l <= 3.9e+150)
		tmp = 2.0 / (2.0 * (((k_m ^ 2.0) * (t_m ^ 3.0)) / (l ^ 2.0)));
	else
		tmp = 2.0 * ((l / sqrt((t_m / (k_m ^ -4.0)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 3.8e-150], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.9e+150], N[(2.0 / N[(2.0 * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[Sqrt[N[(t$95$m / N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 3.7999999999999998e-150

   1. Initial program 54.4%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in t around 0 61.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. associate-/r* 55.9%

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

   7. Simplified 55.9%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 56.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 56.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.9%

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

   11. Simplified 55.9%

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

       1. add-sqr-sqrt 55.9%

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

       2. pow2 55.9%

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

       3. sqrt-prod 55.9%

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

       4. unpow2 55.9%

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

       5. sqrt-prod 18.9%

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

       6. add-sqr-sqrt 62.6%

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

       7. sqrt-pow1 63.4%

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

       8. metadata-eval 63.4%

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

   13. Applied egg-rr 63.4%

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

   if 3.7999999999999998e-150 < l < 3.89999999999999991e150

   1. Initial program 70.0%

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

   2. Taylor expanded in k around 0 60.4%

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

   if 3.89999999999999991e150 < l

   1. Initial program 34.7%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 62.8%

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

   5. Taylor expanded in k around 0 59.5%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 59.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. associate-/l* 59.5%

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

       2. add-sqr-sqrt 21.9%

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

       3. sqrt-div 21.9%

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

       4. unpow2 21.9%

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

       5. sqrt-prod 21.9%

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

       6. add-sqr-sqrt 21.9%

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

       7. sqrt-div 21.9%

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

       8. unpow2 21.9%

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

       9. sqrt-prod 25.3%

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

       10. add-sqr-sqrt 25.3%

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

   13. Applied egg-rr 25.3%

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

       1. unpow2 25.3%

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

   15. Simplified 25.3%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{\frac{t}{{k}^{-4}}}}\right)}^{2}\\ \end{array} \]

Alternative 13: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\\ \mathbf{elif}\;\ell \leq 7.7 \cdot 10^{+149}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k_m}^{2} \cdot {t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{\frac{t_m}{{k_m}^{-4}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 2.9e-150)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (if (<= l 7.7e+149)
      (/ (pow l 2.0) (* (pow k_m 2.0) (pow t_m 3.0)))
      (* 2.0 (pow (/ l (sqrt (/ t_m (pow k_m -4.0)))) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2.9e-150) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else if (l <= 7.7e+149) {
		tmp = pow(l, 2.0) / (pow(k_m, 2.0) * pow(t_m, 3.0));
	} else {
		tmp = 2.0 * pow((l / sqrt((t_m / pow(k_m, -4.0)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2.9d-150) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else if (l <= 7.7d+149) then
        tmp = (l ** 2.0d0) / ((k_m ** 2.0d0) * (t_m ** 3.0d0))
    else
        tmp = 2.0d0 * ((l / sqrt((t_m / (k_m ** (-4.0d0))))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2.9e-150) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else if (l <= 7.7e+149) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k_m, 2.0) * Math.pow(t_m, 3.0));
	} else {
		tmp = 2.0 * Math.pow((l / Math.sqrt((t_m / Math.pow(k_m, -4.0)))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 2.9e-150:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	elif l <= 7.7e+149:
		tmp = math.pow(l, 2.0) / (math.pow(k_m, 2.0) * math.pow(t_m, 3.0))
	else:
		tmp = 2.0 * math.pow((l / math.sqrt((t_m / math.pow(k_m, -4.0)))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 2.9e-150)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	elseif (l <= 7.7e+149)
		tmp = Float64((l ^ 2.0) / Float64((k_m ^ 2.0) * (t_m ^ 3.0)));
	else
		tmp = Float64(2.0 * (Float64(l / sqrt(Float64(t_m / (k_m ^ -4.0)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 2.9e-150)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	elseif (l <= 7.7e+149)
		tmp = (l ^ 2.0) / ((k_m ^ 2.0) * (t_m ^ 3.0));
	else
		tmp = 2.0 * ((l / sqrt((t_m / (k_m ^ -4.0)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 2.9e-150], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.7e+149], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[Sqrt[N[(t$95$m / N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 2.8999999999999998e-150

   1. Initial program 54.4%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in t around 0 61.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. associate-/r* 55.9%

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

   7. Simplified 55.9%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 56.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 56.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.9%

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

   11. Simplified 55.9%

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

       1. add-sqr-sqrt 55.9%

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

       2. pow2 55.9%

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

       3. sqrt-prod 55.9%

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

       4. unpow2 55.9%

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

       5. sqrt-prod 18.9%

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

       6. add-sqr-sqrt 62.6%

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

       7. sqrt-pow1 63.4%

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

       8. metadata-eval 63.4%

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

   13. Applied egg-rr 63.4%

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

   if 2.8999999999999998e-150 < l < 7.69999999999999998e149

   1. Initial program 70.0%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in k around 0 60.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 60.3%

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

   6. Simplified 60.3%

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

   if 7.69999999999999998e149 < l

   1. Initial program 34.7%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 62.8%

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

   5. Taylor expanded in k around 0 59.5%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 59.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. associate-/l* 59.5%

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

       2. add-sqr-sqrt 21.9%

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

       3. sqrt-div 21.9%

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

       4. unpow2 21.9%

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

       5. sqrt-prod 21.9%

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

       6. add-sqr-sqrt 21.9%

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

       7. sqrt-div 21.9%

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

       8. unpow2 21.9%

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

       9. sqrt-prod 25.3%

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

       10. add-sqr-sqrt 25.3%

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

   13. Applied egg-rr 25.3%

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

       1. unpow2 25.3%

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

   15. Simplified 25.3%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 7.7 \cdot 10^{+149}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{\frac{t}{{k}^{-4}}}}\right)}^{2}\\ \end{array} \]

Alternative 14: 57.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot {k_m}^{-2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{t_2}^{2}}{t_m}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k_m}^{2} \cdot {t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{t_2}{t_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* l (pow k_m -2.0))))
   (*
    t_s
    (if (<= l 8.2e-150)
      (* 2.0 (/ (pow t_2 2.0) t_m))
      (if (<= l 4.8e+150)
        (/ (pow l 2.0) (* (pow k_m 2.0) (pow t_m 3.0)))
        (* 2.0 (* t_2 (/ t_2 t_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l * pow(k_m, -2.0);
	double tmp;
	if (l <= 8.2e-150) {
		tmp = 2.0 * (pow(t_2, 2.0) / t_m);
	} else if (l <= 4.8e+150) {
		tmp = pow(l, 2.0) / (pow(k_m, 2.0) * pow(t_m, 3.0));
	} else {
		tmp = 2.0 * (t_2 * (t_2 / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * (k_m ** (-2.0d0))
    if (l <= 8.2d-150) then
        tmp = 2.0d0 * ((t_2 ** 2.0d0) / t_m)
    else if (l <= 4.8d+150) then
        tmp = (l ** 2.0d0) / ((k_m ** 2.0d0) * (t_m ** 3.0d0))
    else
        tmp = 2.0d0 * (t_2 * (t_2 / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l * Math.pow(k_m, -2.0);
	double tmp;
	if (l <= 8.2e-150) {
		tmp = 2.0 * (Math.pow(t_2, 2.0) / t_m);
	} else if (l <= 4.8e+150) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k_m, 2.0) * Math.pow(t_m, 3.0));
	} else {
		tmp = 2.0 * (t_2 * (t_2 / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = l * math.pow(k_m, -2.0)
	tmp = 0
	if l <= 8.2e-150:
		tmp = 2.0 * (math.pow(t_2, 2.0) / t_m)
	elif l <= 4.8e+150:
		tmp = math.pow(l, 2.0) / (math.pow(k_m, 2.0) * math.pow(t_m, 3.0))
	else:
		tmp = 2.0 * (t_2 * (t_2 / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(l * (k_m ^ -2.0))
	tmp = 0.0
	if (l <= 8.2e-150)
		tmp = Float64(2.0 * Float64((t_2 ^ 2.0) / t_m));
	elseif (l <= 4.8e+150)
		tmp = Float64((l ^ 2.0) / Float64((k_m ^ 2.0) * (t_m ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_2 / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = l * (k_m ^ -2.0);
	tmp = 0.0;
	if (l <= 8.2e-150)
		tmp = 2.0 * ((t_2 ^ 2.0) / t_m);
	elseif (l <= 4.8e+150)
		tmp = (l ^ 2.0) / ((k_m ^ 2.0) * (t_m ^ 3.0));
	else
		tmp = 2.0 * (t_2 * (t_2 / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 8.2e-150], N[(2.0 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e+150], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(t$95$2 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 8.1999999999999997e-150

   1. Initial program 54.4%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in t around 0 61.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. associate-/r* 55.9%

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

   7. Simplified 55.9%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 56.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 56.1%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 56.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.9%

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

   11. Simplified 55.9%

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

       1. add-sqr-sqrt 55.9%

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

       2. pow2 55.9%

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

       3. sqrt-prod 55.9%

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

       4. unpow2 55.9%

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

       5. sqrt-prod 18.9%

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

       6. add-sqr-sqrt 62.6%

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

       7. sqrt-pow1 63.4%

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

       8. metadata-eval 63.4%

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

   13. Applied egg-rr 63.4%

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

   if 8.1999999999999997e-150 < l < 4.80000000000000005e150

   1. Initial program 70.0%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in k around 0 60.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. *-commutative 60.3%

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

   6. Simplified 60.3%

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

   if 4.80000000000000005e150 < l

   1. Initial program 34.7%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 62.8%

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

   5. Taylor expanded in k around 0 59.5%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 59.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 59.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. add-sqr-sqrt 59.5%

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

       2. *-un-lft-identity 59.5%

          \[\leadsto 2 \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}{\color{blue}{1 \cdot t}} \]

       3. times-frac 59.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right)} \]

       4. sqrt-prod 59.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       5. unpow2 59.5%

          \[\leadsto 2 \cdot \left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       6. sqrt-prod 59.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       7. add-sqr-sqrt 59.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       8. sqrt-pow1 59.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       9. metadata-eval 59.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{\color{blue}{-2}}}{1} \cdot \frac{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}{t}\right) \]

       10. sqrt-prod 59.5%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{-2}}{1} \cdot \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{t}\right) \]

       11. unpow2 59.5%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{-2}}{1} \cdot \frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{t}\right) \]

       12. sqrt-prod 66.3%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{-2}}{1} \cdot \frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{t}\right) \]

       13. add-sqr-sqrt 66.3%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{-2}}{1} \cdot \frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{t}\right) \]

       14. sqrt-pow1 66.4%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{-2}}{1} \cdot \frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{t}\right) \]

       15. metadata-eval 66.4%

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

   13. Applied egg-rr 66.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot {k}^{-2}}{t}\right)\\ \end{array} \]

Alternative 15: 51.9% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (pow l 2.0) (/ (pow k_m -4.0) t_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Simplified 51.7%

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

4. Taylor expanded in t around 0 61.9%

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

5. Taylor expanded in k around 0 53.0%

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

   1. associate-/r* 53.4%

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

7. Simplified 53.4%

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

   1. expm1-log1p-u 53.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 53.6%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 53.4%

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

11. Simplified 53.4%

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

12. Taylor expanded in l around 0 53.0%

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

    1. *-commutative 53.0%

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

    2. rem-exp-log 41.6%

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

    3. associate-/r* 41.8%

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

    4. log-div 38.8%

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

    5. log-pow 22.6%

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

    6. cancel-sign-sub-inv 22.6%

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

    7. metadata-eval 22.6%

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

    8. log-pow 38.8%

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

    9. log-prod 41.8%

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

    10. associate-/r/ 41.6%

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

    11. rem-exp-log 53.0%

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

    12. associate-/r/ 53.6%

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

    13. *-commutative 53.6%

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

    14. associate-*r/ 53.4%

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

    15. *-commutative 53.4%

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

    16. associate-*r/ 53.0%

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

14. Simplified 53.0%

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

15. Final simplification 53.0%

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

Alternative 16: 52.1% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow l 2.0) (pow k_m -4.0)) t_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Simplified 51.7%

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

4. Taylor expanded in t around 0 61.9%

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

5. Taylor expanded in k around 0 53.0%

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

   1. associate-/r* 53.4%

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

7. Simplified 53.4%

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

   1. expm1-log1p-u 53.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 53.6%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 53.4%

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

11. Simplified 53.4%

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

12. Final simplification 53.4%

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

Alternative 17: 52.6% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k_m 4.0)))))

C

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

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l ** 2.0d0) / t_m) / (k_m ** 4.0d0)))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k_m, 4.0)));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k_m, 4.0)))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k_m ^ 4.0))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l ^ 2.0) / t_m) / (k_m ^ 4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

4. Taylor expanded in t around 0 61.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

5. Taylor expanded in k around 0 53.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. associate-/r* 53.4%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

7. Simplified 53.4%

   \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

8. Taylor expanded in l around 0 53.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. *-commutative 53.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   2. associate-/r* 53.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

10. Simplified 53.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

11. Final simplification 53.6%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]

Alternative 18: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{{\left(\ell \cdot {k_m}^{-2}\right)}^{2}}{t_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m)))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

4. Taylor expanded in t around 0 61.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

5. Taylor expanded in k around 0 53.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. associate-/r* 53.4%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

7. Simplified 53.4%

   \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 53.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 53.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 53.6%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

11. Simplified 53.4%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{t} \]

    2. pow2 53.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\sqrt{{\ell}^{2} \cdot {k}^{-4}}\right)}^{2}}}{t} \]

    3. sqrt-prod 53.4%

       \[\leadsto 2 \cdot \frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}\right)}}^{2}}{t} \]

    4. unpow2 53.4%

       \[\leadsto 2 \cdot \frac{{\left(\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    5. sqrt-prod 32.0%

       \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    6. add-sqr-sqrt 58.3%

       \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    7. sqrt-pow1 58.8%

       \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}\right)}^{2}}{t} \]

    8. metadata-eval 58.8%

       \[\leadsto 2 \cdot \frac{{\left(\ell \cdot {k}^{\color{blue}{-2}}\right)}^{2}}{t} \]

13. Applied egg-rr 58.8%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]

14. Final simplification 58.8%

    \[\leadsto 2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t} \]

Reproduce

herbie shell --seed 2023318 
(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))))