Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 93.0%

Time: 20.5s

Alternatives: 18

Speedup: 1.9×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k \cdot \tan k} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.65e-61)
      (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))
      (if (<= t_m 8.2e+15)
        (* t_2 (* (/ (/ 2.0 (pow t_m 3.0)) (* (sin k) (tan k))) t_2))
        (/
         2.0
         (pow
          (*
           t_m
           (*
            (pow (cbrt l) -2.0)
            (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.65e-61) {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	} else if (t_m <= 8.2e+15) {
		tmp = t_2 * (((2.0 / pow(t_m, 3.0)) / (sin(k) * tan(k))) * t_2);
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.65e-61) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	} else if (t_m <= 8.2e+15) {
		tmp = t_2 * (((2.0 / Math.pow(t_m, 3.0)) / (Math.sin(k) * Math.tan(k))) * t_2);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.65e-61)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	elseif (t_m <= 8.2e+15)
		tmp = Float64(t_2 * Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / Float64(sin(k) * tan(k))) * t_2));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-61], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.2e+15], N[(t$95$2 * N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-61

   1. Initial program 52.2%

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

   3. Simplified 51.6%

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

      1. add-sqr-sqrt 33.4%

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

      2. sqrt-div 32.8%

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

      3. sqrt-div 32.8%

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

   6. Applied egg-rr 32.9%

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

      1. unpow2 32.9%

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

   8. Simplified 33.4%

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

   9. Taylor expanded in t around 0 32.0%

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

       1. *-commutative 32.0%

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

       2. times-frac 32.1%

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

   11. Simplified 32.1%

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

       1. unpow-prod-down 31.5%

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

       2. pow2 31.5%

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

       3. add-sqr-sqrt 74.7%

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

   13. Applied egg-rr 74.7%

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

       1. associate-*l/ 74.7%

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

       2. associate-/l* 74.7%

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

   15. Simplified 74.7%

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

   if 1.64999999999999998e-61 < t < 8.2e15

   1. Initial program 73.1%

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

   3. Simplified 63.2%

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

      1. associate-*r* 69.5%

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

      2. add-sqr-sqrt 69.4%

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

      3. times-frac 79.4%

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

      4. associate-/l/ 79.4%

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

   6. Applied egg-rr 79.4%

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

   8. Simplified 94.4%

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

   if 8.2e15 < t

   1. Initial program 64.7%

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

   3. Simplified 66.7%

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

      1. associate-*l/ 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. associate-*r* 69.6%

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

   8. Simplified 69.6%

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

   10. Applied egg-rr 84.9%

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

       1. unpow2 84.9%

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

       2. unpow3 84.9%

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

       3. associate-*l* 84.9%

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

       4. associate-*r* 84.9%

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

   12. Simplified 84.9%

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

       1. pow1/3 63.1%

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

       2. associate-*l* 63.1%

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

       3. unpow-prod-down 36.6%

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

       4. pow1/3 48.3%

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

   14. Applied egg-rr 48.3%

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

       1. unpow1/3 97.0%

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

   16. Simplified 97.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{t\_2 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<=
         (/ 2.0 (* t_2 (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
         1e+295)
      (/ 2.0 (* t_2 (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
      (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + pow((k / t_m), 2.0));
	double tmp;
	if ((2.0 / (t_2 * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= 1e+295) {
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))
    if ((2.0d0 / (t_2 * (tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))))) <= 1d+295) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l)))))
    else
        tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0d0) / sin(k)) / k)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if ((2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))))) <= 1e+295) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 + (1.0 + math.pow((k / t_m), 2.0))
	tmp = 0
	if (2.0 / (t_2 * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))))) <= 1e+295:
		tmp = 2.0 / (t_2 * (math.tan(k) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))))
	else:
		tmp = (math.cos(k) / t_m) * math.pow((l * ((math.sqrt(2.0) / math.sin(k)) / k)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= 1e+295)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 + (1.0 + ((k / t_m) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / (t_2 * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))))) <= 1e+295)
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l)))));
	else
		tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0) / sin(k)) / k)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+295], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.9999999999999998e294

   1. Initial program 82.2%

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

      1. unpow3 82.2%

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

      2. times-frac 88.6%

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

      3. pow2 88.6%

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

   4. Applied egg-rr 88.6%

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

   if 9.9999999999999998e294 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.5%

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

   3. Simplified 19.5%

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

      1. add-sqr-sqrt 19.5%

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

      2. sqrt-div 19.5%

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

      3. sqrt-div 19.5%

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

   6. Applied egg-rr 30.6%

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

      1. unpow2 30.6%

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

   8. Simplified 39.9%

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

   9. Taylor expanded in t around 0 49.9%

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

       1. *-commutative 49.9%

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

       2. times-frac 50.0%

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

   11. Simplified 50.0%

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

       1. unpow-prod-down 49.1%

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

       2. pow2 49.1%

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

       3. add-sqr-sqrt 79.3%

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

   13. Applied egg-rr 79.3%

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

       1. associate-*l/ 79.3%

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

       2. associate-/l* 79.3%

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

   15. Simplified 79.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t\_2\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+295}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_2))
           (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
         1e+295)
      (* (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) (/ l (+ 2.0 t_2)))
      (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= 1e+295) {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (1.0d0 + t_2)) * (tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))))) <= 1d+295) then
        tmp = (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))) * (l / (2.0d0 + t_2))
    else
        tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0d0) / sin(k)) / k)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))))) <= 1e+295) {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (1.0 + t_2)) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))))) <= 1e+295:
		tmp = (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))) * (l / (2.0 + t_2))
	else:
		tmp = (math.cos(k) / t_m) * math.pow((l * ((math.sqrt(2.0) / math.sin(k)) / k)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_2)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= 1e+295)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + t_2)));
	else
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))))) <= 1e+295)
		tmp = (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))))) * (l / (2.0 + t_2));
	else
		tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0) / sin(k)) / k)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+295], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.9999999999999998e294

   1. Initial program 82.2%

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

   3. Simplified 80.9%

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

      1. associate-*r* 83.5%

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

      2. *-un-lft-identity 83.5%

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

      3. times-frac 85.3%

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

      4. associate-/l/ 85.3%

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

   6. Applied egg-rr 85.3%

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

   if 9.9999999999999998e294 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.5%

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

   3. Simplified 19.5%

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

      1. add-sqr-sqrt 19.5%

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

      2. sqrt-div 19.5%

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

      3. sqrt-div 19.5%

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

   6. Applied egg-rr 30.6%

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

      1. unpow2 30.6%

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

   8. Simplified 39.9%

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

   9. Taylor expanded in t around 0 49.9%

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

       1. *-commutative 49.9%

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

       2. times-frac 50.0%

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

   11. Simplified 50.0%

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

       1. unpow-prod-down 49.1%

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

       2. pow2 49.1%

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

       3. add-sqr-sqrt 79.3%

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

   13. Applied egg-rr 79.3%

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

       1. associate-*l/ 79.3%

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

       2. associate-/l* 79.3%

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

   15. Simplified 79.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+295}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(t\_2 \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(t\_2 \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k 9.5e-69)
      (/ 2.0 (pow (* t_m (* t_2 (* (cbrt (sin k)) (cbrt (* k 2.0))))) 3.0))
      (if (<= k 1.1e+94)
        (/
         2.0
         (pow
          (*
           t_m
           (* t_2 (cbrt (* (* (sin k) (tan k)) (+ 2.0 (pow (/ k t_m) 2.0))))))
          3.0))
        (* (/ (cos k) t_m) (pow (* (/ (sqrt 2.0) (sin k)) (/ l k)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(cbrt(l), -2.0);
	double tmp;
	if (k <= 9.5e-69) {
		tmp = 2.0 / pow((t_m * (t_2 * (cbrt(sin(k)) * cbrt((k * 2.0))))), 3.0);
	} else if (k <= 1.1e+94) {
		tmp = 2.0 / pow((t_m * (t_2 * cbrt(((sin(k) * tan(k)) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	} else {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) / sin(k)) * (l / k)), 2.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k <= 9.5e-69) {
		tmp = 2.0 / Math.pow((t_m * (t_2 * (Math.cbrt(Math.sin(k)) * Math.cbrt((k * 2.0))))), 3.0);
	} else if (k <= 1.1e+94) {
		tmp = 2.0 / Math.pow((t_m * (t_2 * Math.cbrt(((Math.sin(k) * Math.tan(k)) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	} else {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) / Math.sin(k)) * (l / k)), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k <= 9.5e-69)
		tmp = Float64(2.0 / (Float64(t_m * Float64(t_2 * Float64(cbrt(sin(k)) * cbrt(Float64(k * 2.0))))) ^ 3.0));
	elseif (k <= 1.1e+94)
		tmp = Float64(2.0 / (Float64(t_m * Float64(t_2 * cbrt(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	else
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) / sin(k)) * Float64(l / k)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 9.5e-69], N[(2.0 / N[Power[N[(t$95$m * N[(t$95$2 * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e+94], N[(2.0 / N[Power[N[(t$95$m * N[(t$95$2 * N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 9.50000000000000094e-69

   1. Initial program 58.2%

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

   3. Simplified 60.5%

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

      1. associate-*l/ 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-*r* 62.4%

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

   8. Simplified 62.4%

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

   10. Applied egg-rr 78.0%

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

       1. unpow2 78.0%

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

       2. unpow3 78.0%

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

       3. associate-*l* 78.0%

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

       4. associate-*r* 78.0%

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

   12. Simplified 78.0%

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

       1. pow1/3 65.7%

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

       2. associate-*l* 65.7%

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

       3. unpow-prod-down 35.1%

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

       4. pow1/3 44.9%

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

   14. Applied egg-rr 44.9%

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

       1. unpow1/3 89.5%

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

   16. Simplified 89.5%

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

   17. Taylor expanded in k around 0 78.4%

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

       1. *-commutative 78.4%

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

   19. Simplified 78.4%

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

   if 9.50000000000000094e-69 < k < 1.10000000000000006e94

   1. Initial program 61.1%

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

   3. Simplified 66.5%

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

      1. associate-*l/ 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. associate-*r* 68.5%

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

   8. Simplified 68.5%

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

   10. Applied egg-rr 81.0%

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

       1. unpow2 81.0%

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

       2. unpow3 81.0%

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

       3. associate-*l* 81.2%

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

       4. associate-*r* 81.2%

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

   12. Simplified 81.2%

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

   if 1.10000000000000006e94 < k

   1. Initial program 46.9%

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

   3. Simplified 44.6%

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

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-div 42.1%

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

      3. sqrt-div 42.1%

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

   6. Applied egg-rr 39.9%

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

      1. unpow2 39.9%

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

   8. Simplified 44.2%

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

   9. Taylor expanded in t around 0 39.2%

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

       1. *-commutative 39.2%

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

       2. times-frac 39.4%

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

   11. Simplified 39.4%

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

       1. *-commutative 39.4%

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

       2. unpow-prod-down 37.3%

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

       3. pow2 37.3%

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

       4. add-sqr-sqrt 93.0%

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

   13. Applied egg-rr 93.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-68)
    (/
     2.0
     (pow
      (* t_m (* (pow (cbrt l) -2.0) (* (cbrt (sin k)) (cbrt (* k 2.0)))))
      3.0))
    (if (<= k 3.55e+93)
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (* (/ (cos k) t_m) (pow (* (/ (sqrt 2.0) (sin k)) (/ l k)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-68) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(sin(k)) * cbrt((k * 2.0))))), 3.0);
	} else if (k <= 3.55e+93) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) / sin(k)) * (l / k)), 2.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-68) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(Math.sin(k)) * Math.cbrt((k * 2.0))))), 3.0);
	} else if (k <= 3.55e+93) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) / Math.sin(k)) * (l / k)), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-68)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(sin(k)) * cbrt(Float64(k * 2.0))))) ^ 3.0));
	elseif (k <= 3.55e+93)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) / sin(k)) * Float64(l / k)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-68], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.55e+93], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.20000000000000002e-68

   1. Initial program 58.2%

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

   3. Simplified 60.5%

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

      1. associate-*l/ 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-*r* 62.4%

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

   8. Simplified 62.4%

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

   10. Applied egg-rr 78.0%

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

       1. unpow2 78.0%

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

       2. unpow3 78.0%

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

       3. associate-*l* 78.0%

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

       4. associate-*r* 78.0%

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

   12. Simplified 78.0%

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

       1. pow1/3 65.7%

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

       2. associate-*l* 65.7%

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

       3. unpow-prod-down 35.1%

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

       4. pow1/3 44.9%

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

   14. Applied egg-rr 44.9%

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

       1. unpow1/3 89.5%

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

   16. Simplified 89.5%

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

   17. Taylor expanded in k around 0 78.4%

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

       1. *-commutative 78.4%

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

   19. Simplified 78.4%

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

   if 2.20000000000000002e-68 < k < 3.5500000000000002e93

   1. Initial program 61.1%

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

      1. add-sqr-sqrt 29.0%

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

      2. pow2 29.0%

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

      3. sqrt-div 29.0%

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

      4. sqrt-pow1 29.0%

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

      5. metadata-eval 29.0%

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

      6. sqrt-prod 13.1%

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

      7. add-sqr-sqrt 34.1%

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

   4. Applied egg-rr 34.1%

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

   if 3.5500000000000002e93 < k

   1. Initial program 46.9%

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

   3. Simplified 44.6%

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

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-div 42.1%

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

      3. sqrt-div 42.1%

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

   6. Applied egg-rr 39.9%

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

      1. unpow2 39.9%

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

   8. Simplified 44.2%

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

   9. Taylor expanded in t around 0 39.2%

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

       1. *-commutative 39.2%

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

       2. times-frac 39.4%

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

   11. Simplified 39.4%

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

       1. *-commutative 39.4%

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

       2. unpow-prod-down 37.3%

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

       3. pow2 37.3%

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

       4. add-sqr-sqrt 93.0%

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

   13. Applied egg-rr 93.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 20000000000:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + t\_2\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 20000000000.0)
      (* (/ (cos k) t_m) (pow (* (/ (sqrt 2.0) (sin k)) (/ l k)) 2.0))
      (if (<= t_m 5.2e+102)
        (*
         (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))
         (/ l (+ 2.0 t_2)))
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
          (+ 1.0 (+ 1.0 t_2)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 20000000000.0) {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) / sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 5.2e+102) {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 20000000000.0d0) then
        tmp = (cos(k) / t_m) * (((sqrt(2.0d0) / sin(k)) * (l / k)) ** 2.0d0)
    else if (t_m <= 5.2d+102) then
        tmp = (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))) * (l / (2.0d0 + t_2))
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (1.0d0 + (1.0d0 + t_2)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 20000000000.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) / Math.sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 5.2e+102) {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 20000000000.0:
		tmp = (math.cos(k) / t_m) * math.pow(((math.sqrt(2.0) / math.sin(k)) * (l / k)), 2.0)
	elif t_m <= 5.2e+102:
		tmp = (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))) * (l / (2.0 + t_2))
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 20000000000.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) / sin(k)) * Float64(l / k)) ^ 2.0));
	elseif (t_m <= 5.2e+102)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(1.0 + t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 20000000000.0)
		tmp = (cos(k) / t_m) * (((sqrt(2.0) / sin(k)) * (l / k)) ^ 2.0);
	elseif (t_m <= 5.2e+102)
		tmp = (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))))) * (l / (2.0 + t_2));
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (1.0 + (1.0 + t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 20000000000.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e+102], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2e10

   1. Initial program 53.8%

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

   3. Simplified 52.2%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-div 33.5%

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

      3. sqrt-div 33.5%

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

   6. Applied egg-rr 34.7%

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

      1. unpow2 34.7%

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

   8. Simplified 36.7%

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

   9. Taylor expanded in t around 0 33.5%

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

       1. *-commutative 33.5%

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

       2. times-frac 33.5%

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

   11. Simplified 33.5%

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

       1. *-commutative 33.5%

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

       2. unpow-prod-down 33.1%

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

       3. pow2 33.1%

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

       4. add-sqr-sqrt 74.5%

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

   13. Applied egg-rr 74.5%

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

   if 2e10 < t < 5.20000000000000013e102

   1. Initial program 74.0%

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

   3. Simplified 78.8%

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

      1. associate-*r* 89.3%

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

      2. *-un-lft-identity 89.3%

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

      3. times-frac 94.3%

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

      4. associate-/l/ 94.4%

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

   6. Applied egg-rr 94.4%

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

   if 5.20000000000000013e102 < t

   1. Initial program 62.6%

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

      1. add-sqr-sqrt 62.6%

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

      2. pow2 62.6%

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

      3. sqrt-div 62.6%

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

      4. sqrt-pow1 71.6%

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

      5. metadata-eval 71.6%

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

      6. sqrt-prod 40.3%

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

      7. add-sqr-sqrt 89.1%

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

   4. Applied egg-rr 89.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 20000000000:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 51000000000:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 3.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell \cdot 2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 51000000000.0)
    (* (/ (cos k) t_m) (pow (* (/ (sqrt 2.0) (sin k)) (/ l k)) 2.0))
    (if (<= t_m 3.8e+93)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ (* l 2.0) (* (pow t_m 3.0) (* (sin k) (tan k)))))
      (/
       2.0
       (/ (pow (* (/ t_m (cbrt l)) (cbrt (* 2.0 (pow k 2.0)))) 3.0) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 51000000000.0) {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) / sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 3.8e+93) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * ((l * 2.0) / (pow(t_m, 3.0) * (sin(k) * tan(k))));
	} else {
		tmp = 2.0 / (pow(((t_m / cbrt(l)) * cbrt((2.0 * pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 51000000000.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) / Math.sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 3.8e+93) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l * 2.0) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 51000000000.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) / sin(k)) * Float64(l / k)) ^ 2.0));
	elseif (t_m <= 3.8e+93)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l * 2.0) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 51000000000.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.8e+93], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.1e10

   1. Initial program 53.8%

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

   3. Simplified 52.2%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-div 33.5%

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

      3. sqrt-div 33.5%

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

   6. Applied egg-rr 34.7%

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

      1. unpow2 34.7%

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

   8. Simplified 36.7%

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

   9. Taylor expanded in t around 0 33.5%

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

       1. *-commutative 33.5%

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

       2. times-frac 33.5%

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

   11. Simplified 33.5%

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

       1. *-commutative 33.5%

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

       2. unpow-prod-down 33.1%

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

       3. pow2 33.1%

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

       4. add-sqr-sqrt 74.5%

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

   13. Applied egg-rr 74.5%

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

   if 5.1e10 < t < 3.7999999999999998e93

   1. Initial program 76.6%

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

   3. Simplified 81.9%

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

      1. associate-*r* 88.1%

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

      2. *-un-lft-identity 88.1%

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

      3. times-frac 93.6%

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

      4. associate-/l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. /-rgt-identity 93.8%

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

      2. associate-*l/ 93.8%

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

      3. associate-*l* 81.9%

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

   8. Simplified 81.9%

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

   if 3.7999999999999998e93 < t

   1. Initial program 62.2%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. associate-*l/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. cbrt-prod 66.7%

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

      4. cbrt-div 66.7%

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

      5. unpow3 66.7%

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

      6. add-cbrt-cube 79.3%

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

   9. Applied egg-rr 79.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 51000000000:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell \cdot 2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 17500000000:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{+93}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{{t\_m}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 17500000000.0)
    (* (/ (cos k) t_m) (pow (* (/ (sqrt 2.0) (sin k)) (/ l k)) 2.0))
    (if (<= t_m 2.2e+93)
      (*
       l
       (/
        2.0
        (*
         (+ 2.0 (pow (/ k t_m) 2.0))
         (* (* (sin k) (tan k)) (/ (pow t_m 3.0) l)))))
      (/
       2.0
       (/ (pow (* (/ t_m (cbrt l)) (cbrt (* 2.0 (pow k 2.0)))) 3.0) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 17500000000.0) {
		tmp = (cos(k) / t_m) * pow(((sqrt(2.0) / sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 2.2e+93) {
		tmp = l * (2.0 / ((2.0 + pow((k / t_m), 2.0)) * ((sin(k) * tan(k)) * (pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (pow(((t_m / cbrt(l)) * cbrt((2.0 * pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 17500000000.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow(((Math.sqrt(2.0) / Math.sin(k)) * (l / k)), 2.0);
	} else if (t_m <= 2.2e+93) {
		tmp = l * (2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * ((Math.sin(k) * Math.tan(k)) * (Math.pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 17500000000.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(Float64(sqrt(2.0) / sin(k)) * Float64(l / k)) ^ 2.0));
	elseif (t_m <= 2.2e+93)
		tmp = Float64(l * Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(Float64(sin(k) * tan(k)) * Float64((t_m ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 17500000000.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+93], N[(l * N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.75e10

   1. Initial program 53.8%

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

   3. Simplified 52.2%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-div 33.5%

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

      3. sqrt-div 33.5%

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

   6. Applied egg-rr 34.7%

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

      1. unpow2 34.7%

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

   8. Simplified 36.7%

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

   9. Taylor expanded in t around 0 33.5%

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

       1. *-commutative 33.5%

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

       2. times-frac 33.5%

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

   11. Simplified 33.5%

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

       1. *-commutative 33.5%

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

       2. unpow-prod-down 33.1%

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

       3. pow2 33.1%

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

       4. add-sqr-sqrt 74.5%

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

   13. Applied egg-rr 74.5%

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

   if 1.75e10 < t < 2.20000000000000021e93

   1. Initial program 76.6%

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

   3. Simplified 70.7%

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

      1. associate-*l/ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. associate-*r* 82.0%

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

   8. Simplified 82.0%

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

      1. associate-/r/ 81.8%

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

   10. Applied egg-rr 81.8%

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

   if 2.20000000000000021e93 < t

   1. Initial program 62.2%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. associate-*l/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. cbrt-prod 66.7%

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

      4. cbrt-div 66.7%

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

      5. unpow3 66.7%

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

      6. add-cbrt-cube 79.3%

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

   9. Applied egg-rr 79.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 17500000000:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\ell}{k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+93}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{+89}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e+89)
    (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))
    (/ 2.0 (/ (pow (* (/ t_m (cbrt l)) (cbrt (* 2.0 (pow k 2.0)))) 3.0) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e+89) {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / (pow(((t_m / cbrt(l)) * cbrt((2.0 * pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e+89) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e+89)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e+89], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9e89

   1. Initial program 55.5%

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

   3. Simplified 54.5%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-div 36.7%

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

      3. sqrt-div 36.7%

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

   6. Applied egg-rr 36.8%

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

      1. unpow2 36.8%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in t around 0 33.5%

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

       1. *-commutative 33.5%

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

       2. times-frac 33.5%

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

   11. Simplified 33.5%

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

       1. unpow-prod-down 33.1%

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

       2. pow2 33.1%

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

       3. add-sqr-sqrt 73.3%

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

   13. Applied egg-rr 73.3%

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

       1. associate-*l/ 73.3%

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

       2. associate-/l* 73.3%

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

   15. Simplified 73.3%

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

   if 9e89 < t

   1. Initial program 62.2%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. associate-*l/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. cbrt-prod 66.7%

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

      4. cbrt-div 66.7%

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

      5. unpow3 66.7%

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

      6. add-cbrt-cube 79.3%

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

   9. Applied egg-rr 79.3%

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

Alternative 10: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e+101)
    (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (pow (* t_m (pow (cbrt l) -2.0)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+101) {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+101) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e+101)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e+101], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.4999999999999995e101

   1. Initial program 55.5%

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

   3. Simplified 54.5%

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

      1. add-sqr-sqrt 38.3%

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

      2. sqrt-div 36.9%

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

      3. sqrt-div 36.9%

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

   6. Applied egg-rr 37.0%

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

      1. unpow2 37.0%

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

   8. Simplified 37.8%

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

   9. Taylor expanded in t around 0 33.2%

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

       1. *-commutative 33.2%

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

       2. times-frac 33.2%

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

   11. Simplified 33.2%

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

       1. unpow-prod-down 32.8%

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

       2. pow2 32.8%

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

       3. add-sqr-sqrt 73.1%

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

   13. Applied egg-rr 73.1%

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

       1. associate-*l/ 73.1%

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

       2. associate-/l* 73.1%

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

   15. Simplified 73.1%

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

   if 7.4999999999999995e101 < t

   1. Initial program 62.6%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. add-cube-cbrt 67.4%

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

      2. pow3 67.4%

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

      3. associate-/l/ 58.0%

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

      4. pow2 58.0%

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

      5. cbrt-div 58.0%

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

      6. unpow3 58.0%

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

      7. add-cbrt-cube 67.1%

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

      8. pow2 67.1%

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

      9. cbrt-unprod 80.3%

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

      10. unpow2 80.3%

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

      11. div-inv 80.3%

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

      12. unpow-prod-down 58.3%

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

      13. pow-flip 58.3%

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

      14. metadata-eval 58.3%

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

   7. Applied egg-rr 58.3%

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

      1. cube-prod 80.3%

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

   9. Simplified 80.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{1.5} \cdot \frac{\frac{{t\_m}^{1.5}}{\ell}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e+101)
    (* (/ (cos k) t_m) (pow (* l (/ (/ (sqrt 2.0) (sin k)) k)) 2.0))
    (/
     2.0
     (* (* 2.0 (pow k 2.0)) (* (pow t_m 1.5) (/ (/ (pow t_m 1.5) l) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e+101) {
		tmp = (cos(k) / t_m) * pow((l * ((sqrt(2.0) / sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 1.5) * ((pow(t_m, 1.5) / l) / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5d+101) then
        tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0d0) / sin(k)) / k)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m ** 1.5d0) * (((t_m ** 1.5d0) / l) / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e+101) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * ((Math.sqrt(2.0) / Math.sin(k)) / k)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 1.5) * ((Math.pow(t_m, 1.5) / l) / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5e+101:
		tmp = (math.cos(k) / t_m) * math.pow((l * ((math.sqrt(2.0) / math.sin(k)) / k)), 2.0)
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 1.5) * ((math.pow(t_m, 1.5) / l) / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e+101)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(Float64(sqrt(2.0) / sin(k)) / k)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 1.5) * Float64(Float64((t_m ^ 1.5) / l) / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5e+101)
		tmp = (cos(k) / t_m) * ((l * ((sqrt(2.0) / sin(k)) / k)) ^ 2.0);
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m ^ 1.5) * (((t_m ^ 1.5) / l) / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e+101], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.99999999999999989e101

   1. Initial program 55.5%

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

   3. Simplified 54.5%

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

      1. add-sqr-sqrt 38.3%

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

      2. sqrt-div 36.9%

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

      3. sqrt-div 36.9%

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

   6. Applied egg-rr 37.0%

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

      1. unpow2 37.0%

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

   8. Simplified 37.8%

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

   9. Taylor expanded in t around 0 33.2%

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

       1. *-commutative 33.2%

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

       2. times-frac 33.2%

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

   11. Simplified 33.2%

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

       1. unpow-prod-down 32.8%

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

       2. pow2 32.8%

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

       3. add-sqr-sqrt 73.1%

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

   13. Applied egg-rr 73.1%

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

       1. associate-*l/ 73.1%

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

       2. associate-/l* 73.1%

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

   15. Simplified 73.1%

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

   if 4.99999999999999989e101 < t

   1. Initial program 62.6%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. metadata-eval 67.4%

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

      2. pow-prod-up 67.4%

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

      3. associate-*r/ 76.1%

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

      4. associate-/l* 78.4%

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

   7. Applied egg-rr 78.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{1.5} \cdot \frac{\frac{{t}^{1.5}}{\ell}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{1.5} \cdot \frac{\frac{{t\_m}^{1.5}}{\ell}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-46)
    (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (/
     2.0
     (* (* 2.0 (pow k 2.0)) (* (pow t_m 1.5) (/ (/ (pow t_m 1.5) l) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-46) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 1.5) * ((pow(t_m, 1.5) / l) / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.8d-46) then
        tmp = (((l * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m ** 1.5d0) * (((t_m ** 1.5d0) / l) / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-46) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 1.5) * ((Math.pow(t_m, 1.5) / l) / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.8e-46:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 1.5) * ((math.pow(t_m, 1.5) / l) / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-46)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 1.5) * Float64(Float64((t_m ^ 1.5) / l) / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.8e-46)
		tmp = (((l * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m ^ 1.5) * (((t_m ^ 1.5) / l) / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-46], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8e-46

   1. Initial program 52.2%

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

   3. Simplified 51.6%

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

      1. add-sqr-sqrt 33.8%

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

      2. sqrt-div 33.2%

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

      3. sqrt-div 33.2%

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

   6. Applied egg-rr 33.8%

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

      1. unpow2 33.8%

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

   8. Simplified 34.9%

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

   9. Taylor expanded in t around 0 33.0%

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

       1. *-commutative 33.0%

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

       2. times-frac 33.0%

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

   11. Simplified 33.0%

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

   12. Taylor expanded in k around 0 18.6%

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

   if 1.8e-46 < t

   1. Initial program 67.6%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in k around 0 64.3%

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

      1. metadata-eval 64.3%

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

      2. pow-prod-up 64.3%

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

      3. associate-*r/ 69.5%

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

      4. associate-/l* 70.8%

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

   7. Applied egg-rr 70.8%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{1.5} \cdot \frac{\frac{{t}^{1.5}}{\ell}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{1.5} \cdot \frac{\frac{{t\_m}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.2e+111)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (* (pow t_m 1.5) (/ (/ (pow t_m 1.5) l) l))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.2e+111) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 1.5) * ((pow(t_m, 1.5) / l) / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.2d+111) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m ** 1.5d0) * (((t_m ** 1.5d0) / l) / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.2e+111) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 1.5) * ((Math.pow(t_m, 1.5) / l) / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.2e+111:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 1.5) * ((math.pow(t_m, 1.5) / l) / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.2e+111)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 1.5) * Float64(Float64((t_m ^ 1.5) / l) / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.2e+111)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m ^ 1.5) * (((t_m ^ 1.5) / l) / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.2e+111], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000004e111

   1. Initial program 57.7%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in k around 0 58.3%

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

      1. metadata-eval 58.3%

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

      2. pow-prod-up 29.6%

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

      3. associate-*r/ 31.9%

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

      4. associate-/l* 32.9%

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

   7. Applied egg-rr 32.9%

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

   if 7.2000000000000004e111 < k

   1. Initial program 51.4%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 48.6%

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

      2. sqrt-div 48.6%

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

      3. sqrt-div 48.6%

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

   6. Applied egg-rr 41.0%

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

      1. unpow2 41.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in t around 0 40.3%

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

       1. *-commutative 40.3%

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

       2. times-frac 40.4%

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

   11. Simplified 40.4%

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

   12. Taylor expanded in k around 0 57.8%

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

       1. *-commutative 57.8%

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

       2. unpow2 57.8%

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

       3. rem-square-sqrt 57.8%

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

       4. *-commutative 57.8%

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

   14. Simplified 57.8%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{1.5} \cdot \frac{\frac{{t}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.5e+111)
    (/ 2.0 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (pow k 2.0)) l)))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.5e+111) {
		tmp = 2.0 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.5d+111) then
        tmp = 2.0d0 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k ** 2.0d0)) / l))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.5e+111) {
		tmp = 2.0 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.5e+111:
		tmp = 2.0 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * math.pow(k, 2.0)) / l))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.5e+111)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.5e+111)
		tmp = 2.0 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k ^ 2.0)) / l));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.5e+111], N[(2.0 / N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999948e111

   1. Initial program 57.7%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in k around 0 58.3%

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

      1. associate-*l/ 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. associate-/l* 59.1%

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

   9. Simplified 59.1%

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

       1. cube-mult 59.1%

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

       2. *-un-lft-identity 59.1%

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

       3. times-frac 61.0%

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

       4. pow2 61.0%

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

   11. Applied egg-rr 61.0%

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

   if 7.49999999999999948e111 < k

   1. Initial program 51.4%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 48.6%

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

      2. sqrt-div 48.6%

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

      3. sqrt-div 48.6%

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

   6. Applied egg-rr 41.0%

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

      1. unpow2 41.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in t around 0 40.3%

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

       1. *-commutative 40.3%

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

       2. times-frac 40.4%

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

   11. Simplified 40.4%

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

   12. Taylor expanded in k around 0 57.8%

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

       1. *-commutative 57.8%

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

       2. unpow2 57.8%

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

       3. rem-square-sqrt 57.8%

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

       4. *-commutative 57.8%

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

   14. Simplified 57.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \left({t\_m}^{3} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.7e+111)
    (/ 2.0 (* (/ (* 2.0 (pow k 2.0)) l) (* (pow t_m 3.0) (/ 1.0 l))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.7e+111) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / l) * (pow(t_m, 3.0) * (1.0 / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.7d+111) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) * (1.0d0 / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.7e+111) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) * (1.0 / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.7e+111:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / l) * (math.pow(t_m, 3.0) * (1.0 / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.7e+111)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((t_m ^ 3.0) * Float64(1.0 / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.7e+111)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) * (1.0 / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.7e+111], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.70000000000000008e111

   1. Initial program 57.7%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in k around 0 58.3%

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

      1. associate-*l/ 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. associate-/l* 59.1%

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

   9. Simplified 59.1%

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

       1. div-inv 59.1%

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

   11. Applied egg-rr 59.1%

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

   if 4.70000000000000008e111 < k

   1. Initial program 51.4%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 48.6%

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

      2. sqrt-div 48.6%

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

      3. sqrt-div 48.6%

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

   6. Applied egg-rr 41.0%

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

      1. unpow2 41.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in t around 0 40.3%

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

       1. *-commutative 40.3%

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

       2. times-frac 40.4%

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

   11. Simplified 40.4%

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

   12. Taylor expanded in k around 0 57.8%

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

       1. *-commutative 57.8%

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

       2. unpow2 57.8%

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

       3. rem-square-sqrt 57.8%

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

       4. *-commutative 57.8%

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

   14. Simplified 57.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \left({t}^{3} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.4e+111)
    (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k 2.0)) l)))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.4e+111) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.4d+111) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((2.0d0 * (k ** 2.0d0)) / l))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.4e+111) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.4e+111:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((2.0 * math.pow(k, 2.0)) / l))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.4e+111)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.4e+111)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((2.0 * (k ^ 2.0)) / l));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.4e+111], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.4000000000000005e111

   1. Initial program 57.7%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in k around 0 58.3%

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

      1. associate-*l/ 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. associate-/l* 59.1%

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

   9. Simplified 59.1%

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

   if 7.4000000000000005e111 < k

   1. Initial program 51.4%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 48.6%

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

      2. sqrt-div 48.6%

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

      3. sqrt-div 48.6%

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

   6. Applied egg-rr 41.0%

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

      1. unpow2 41.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in t around 0 40.3%

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

       1. *-commutative 40.3%

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

       2. times-frac 40.4%

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

   11. Simplified 40.4%

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

   12. Taylor expanded in k around 0 57.8%

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

       1. *-commutative 57.8%

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

       2. unpow2 57.8%

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

       3. rem-square-sqrt 57.8%

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

       4. *-commutative 57.8%

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

   14. Simplified 57.8%

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

Alternative 17: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{+36}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7e+36)
    (* l (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow t_m 3.0) l))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7e+36) {
		tmp = l * (2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 3.0) / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d+36) then
        tmp = l * (2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m ** 3.0d0) / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7e+36) {
		tmp = l * (2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 3.0) / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7e+36:
		tmp = l * (2.0 / ((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 3.0) / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7e+36)
		tmp = Float64(l * Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 3.0) / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7e+36)
		tmp = l * (2.0 / ((2.0 * (k ^ 2.0)) * ((t_m ^ 3.0) / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7e+36], N[(l * N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999996e36

   1. Initial program 59.1%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in k around 0 60.2%

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

      1. associate-*l/ 61.0%

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

   7. Applied egg-rr 61.0%

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

      1. associate-/r/ 61.0%

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

   9. Applied egg-rr 61.0%

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

   if 6.9999999999999996e36 < k

   1. Initial program 47.6%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 41.9%

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

      2. sqrt-div 38.1%

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

      3. sqrt-div 38.1%

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

   6. Applied egg-rr 34.5%

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

      1. unpow2 34.5%

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

   8. Simplified 40.0%

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

   9. Taylor expanded in t around 0 37.6%

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

       1. *-commutative 37.6%

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

       2. times-frac 37.7%

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

   11. Simplified 37.7%

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

   12. Taylor expanded in k around 0 52.1%

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

       1. *-commutative 52.1%

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

       2. unpow2 52.1%

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

       3. rem-square-sqrt 52.1%

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

       4. *-commutative 52.1%

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

   14. Simplified 52.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{+36}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

3. Simplified 55.9%

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

   1. add-sqr-sqrt 42.6%

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

   2. sqrt-div 41.4%

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

   3. sqrt-div 41.4%

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

6. Applied egg-rr 43.1%

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

   1. unpow2 43.1%

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

8. Simplified 43.0%

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

9. Taylor expanded in t around 0 32.9%

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

    1. *-commutative 32.9%

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

    2. times-frac 33.0%

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

11. Simplified 33.0%

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

12. Taylor expanded in k around 0 48.4%

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

    1. *-commutative 48.4%

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

    2. unpow2 48.4%

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

    3. rem-square-sqrt 48.4%

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

    4. *-commutative 48.4%

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

14. Simplified 48.4%

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

Reproduce

herbie shell --seed 2024132 
(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))))