Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 79.4%

Time: 41.7s

Alternatives: 20

Speedup: 2.0×

• 26.7% 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: 55.3% 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: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t\_m}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+81}:\\ \;\;\;\;t\_2 \cdot \left(t\_2 \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_2 \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t\_m \cdot \sqrt[3]{k}\right)\right)}^{3}}\right)\\ \end{array} \end{array} \end{array} \]

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 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 4.3e-123)
      (/
       2.0
       (* (pow k 2.0) (* (/ t_m (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
      (if (<= t_m 2.9e+81)
        (* t_2 (* t_2 (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (*
         t_2
         (* t_2 (/ 2.0 (pow (* (cbrt (tan k)) (* t_m (cbrt k))) 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 <= 4.3e-123) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else if (t_m <= 2.9e+81) {
		tmp = t_2 * (t_2 * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = t_2 * (t_2 * (2.0 / pow((cbrt(tan(k)) * (t_m * cbrt(k))), 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 <= 4.3e-123) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else if (t_m <= 2.9e+81) {
		tmp = t_2 * (t_2 * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = t_2 * (t_2 * (2.0 / Math.pow((Math.cbrt(Math.tan(k)) * (t_m * Math.cbrt(k))), 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 <= 4.3e-123)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	elseif (t_m <= 2.9e+81)
		tmp = Float64(t_2 * Float64(t_2 * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(t_2 * Float64(t_2 * Float64(2.0 / (Float64(cbrt(tan(k)) * Float64(t_m * cbrt(k))) ^ 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, 4.3e-123], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+81], N[(t$95$2 * N[(t$95$2 * 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]), $MachinePrecision], N[(t$95$2 * N[(t$95$2 * N[(2.0 / N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.30000000000000032e-123

   1. Initial program 56.8%

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

   3. Simplified 56.6%

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

      1. associate-/r* 52.3%

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

      2. unpow3 52.3%

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

      3. times-frac 59.7%

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

      4. pow2 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in t around 0 61.6%

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

      1. associate-/l* 63.5%

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

      2. times-frac 65.4%

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

   9. Simplified 65.4%

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

   if 4.30000000000000032e-123 < t < 2.9e81

   1. Initial program 61.3%

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

   3. Simplified 61.2%

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

      1. associate-*r* 69.0%

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

      2. add-sqr-sqrt 69.0%

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

      3. times-frac 69.1%

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

   6. Applied egg-rr 77.0%

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

      1. associate-/l* 84.4%

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

      2. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if 2.9e81 < t

   1. Initial program 57.5%

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

   3. Simplified 49.4%

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

      1. associate-*r* 56.0%

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

      2. add-sqr-sqrt 56.0%

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

      3. times-frac 56.0%

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

   6. Applied egg-rr 64.2%

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

      1. associate-/l* 64.2%

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

      2. *-commutative 64.2%

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

   8. Simplified 64.2%

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

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.2%

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

      3. *-commutative 64.2%

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

      4. cbrt-prod 64.2%

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

      5. *-commutative 64.2%

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

      6. cbrt-prod 64.2%

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

      7. unpow3 64.2%

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

      8. add-cbrt-cube 86.6%

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

   10. Applied egg-rr 86.6%

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

   11. Taylor expanded in k around 0 86.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t\_m}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t\_m \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot t\_2\right)\\ \end{array} \end{array} \end{array} \]

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 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 4.3e-123)
      (/
       2.0
       (* (pow k 2.0) (* (/ t_m (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
      (*
       t_2
       (* (/ 2.0 (pow (* (cbrt (tan k)) (* t_m (cbrt (sin k)))) 3.0)) 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 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 4.3e-123) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else {
		tmp = t_2 * ((2.0 / pow((cbrt(tan(k)) * (t_m * cbrt(sin(k)))), 3.0)) * t_2);
	}
	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 <= 4.3e-123) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else {
		tmp = t_2 * ((2.0 / Math.pow((Math.cbrt(Math.tan(k)) * (t_m * Math.cbrt(Math.sin(k)))), 3.0)) * 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(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 4.3e-123)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	else
		tmp = Float64(t_2 * Float64(Float64(2.0 / (Float64(cbrt(tan(k)) * Float64(t_m * cbrt(sin(k)))) ^ 3.0)) * t_2));
	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, 4.3e-123], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(2.0 / N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.30000000000000032e-123

   1. Initial program 56.8%

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

   3. Simplified 56.6%

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

      1. associate-/r* 52.3%

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

      2. unpow3 52.3%

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

      3. times-frac 59.7%

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

      4. pow2 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in t around 0 61.6%

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

      1. associate-/l* 63.5%

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

      2. times-frac 65.4%

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

   9. Simplified 65.4%

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

   if 4.30000000000000032e-123 < t

   1. Initial program 59.4%

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

   3. Simplified 55.3%

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

      1. associate-*r* 62.5%

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

      2. add-sqr-sqrt 62.5%

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

      3. times-frac 62.5%

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

   6. Applied egg-rr 70.6%

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

      1. associate-/l* 74.3%

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

      2. *-commutative 74.3%

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

   8. Simplified 74.3%

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

      1. add-cube-cbrt 74.0%

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

      2. pow3 74.0%

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

      3. *-commutative 74.0%

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

      4. cbrt-prod 74.0%

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

      5. *-commutative 74.0%

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

      6. cbrt-prod 73.9%

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

      7. unpow3 73.9%

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

      8. add-cbrt-cube 85.1%

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

   10. Applied egg-rr 85.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t\_m}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+90}:\\ \;\;\;\;t\_2 \cdot \left(t\_2 \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t\_m \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot t\_2\right) \cdot \left(\ell \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \end{array} \]

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 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 4.3e-123)
      (/
       2.0
       (* (pow k 2.0) (* (/ t_m (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
      (if (<= t_m 7.4e+90)
        (* t_2 (* t_2 (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (*
         (* (/ 2.0 (pow (* (cbrt (tan k)) (* t_m (cbrt (sin k)))) 3.0)) t_2)
         (* l (sqrt 0.5))))))))

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 <= 4.3e-123) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else if (t_m <= 7.4e+90) {
		tmp = t_2 * (t_2 * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = ((2.0 / pow((cbrt(tan(k)) * (t_m * cbrt(sin(k)))), 3.0)) * t_2) * (l * sqrt(0.5));
	}
	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 <= 4.3e-123) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else if (t_m <= 7.4e+90) {
		tmp = t_2 * (t_2 * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = ((2.0 / Math.pow((Math.cbrt(Math.tan(k)) * (t_m * Math.cbrt(Math.sin(k)))), 3.0)) * t_2) * (l * Math.sqrt(0.5));
	}
	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 <= 4.3e-123)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	elseif (t_m <= 7.4e+90)
		tmp = Float64(t_2 * Float64(t_2 * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(cbrt(tan(k)) * Float64(t_m * cbrt(sin(k)))) ^ 3.0)) * t_2) * Float64(l * sqrt(0.5)));
	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, 4.3e-123], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+90], N[(t$95$2 * N[(t$95$2 * 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]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.30000000000000032e-123

   1. Initial program 56.8%

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

   3. Simplified 56.6%

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

      1. associate-/r* 52.3%

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

      2. unpow3 52.3%

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

      3. times-frac 59.7%

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

      4. pow2 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in t around 0 61.6%

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

      1. associate-/l* 63.5%

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

      2. times-frac 65.4%

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

   9. Simplified 65.4%

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

   if 4.30000000000000032e-123 < t < 7.4e90

   1. Initial program 62.8%

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

   3. Simplified 62.6%

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

      1. associate-*r* 70.2%

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

      2. add-sqr-sqrt 70.2%

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

      3. times-frac 70.3%

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

   6. Applied egg-rr 77.9%

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

      1. associate-/l* 85.0%

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

      2. *-commutative 85.0%

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

   8. Simplified 85.0%

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

   if 7.4e90 < t

   1. Initial program 55.8%

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

   3. Simplified 47.3%

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

      1. associate-*r* 54.2%

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

      2. add-sqr-sqrt 54.2%

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

      3. times-frac 54.2%

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

   6. Applied egg-rr 62.7%

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

      1. associate-/l* 62.7%

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

      2. *-commutative 62.7%

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

   8. Simplified 62.7%

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

      1. add-cube-cbrt 62.7%

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

      2. pow3 62.7%

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

      3. *-commutative 62.7%

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

      4. cbrt-prod 62.7%

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

      5. *-commutative 62.7%

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

      6. cbrt-prod 62.7%

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

      7. unpow3 62.7%

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

      8. add-cbrt-cube 86.1%

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

   10. Applied egg-rr 86.1%

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

   11. Taylor expanded in k around 0 86.0%

       \[\leadsto \left(\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \color{blue}{\left(\ell \cdot \sqrt{0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \left(\ell \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+24}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{t\_m}^{1.5}}{\ell} \cdot \left(t\_2 \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t\_m}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

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 (hypot 1.0 (hypot 1.0 (/ k t_m)))))
   (*
    t_s
    (if (<= k 1.25e-154)
      (/
       2.0
       (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
      (if (<= k 5e+24)
        (pow
         (/
          (sqrt 2.0)
          (* (/ (pow t_m 1.5) l) (* t_2 (sqrt (* (sin k) (tan k))))))
         2.0)
        (*
         (/ l t_2)
         (*
          2.0
          (/ (* (cos k) (/ l k)) (* (pow (sin k) 2.0) (pow t_m 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (k <= 1.25e-154) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 5e+24) {
		tmp = pow((sqrt(2.0) / ((pow(t_m, 1.5) / l) * (t_2 * sqrt((sin(k) * tan(k)))))), 2.0);
	} else {
		tmp = (l / t_2) * (2.0 * ((cos(k) * (l / k)) / (pow(sin(k), 2.0) * pow(t_m, 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.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (k <= 1.25e-154) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 5e+24) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(t_m, 1.5) / l) * (t_2 * Math.sqrt((Math.sin(k) * Math.tan(k)))))), 2.0);
	} else {
		tmp = (l / t_2) * (2.0 * ((Math.cos(k) * (l / k)) / (Math.pow(Math.sin(k), 2.0) * Math.pow(t_m, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.hypot(1.0, math.hypot(1.0, (k / t_m)))
	tmp = 0
	if k <= 1.25e-154:
		tmp = 2.0 / (math.pow(((math.pow(t_m, 1.5) * (1.0 / l)) * math.sqrt(math.sin(k))), 2.0) * (2.0 * k))
	elif k <= 5e+24:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(t_m, 1.5) / l) * (t_2 * math.sqrt((math.sin(k) * math.tan(k)))))), 2.0)
	else:
		tmp = (l / t_2) * (2.0 * ((math.cos(k) * (l / k)) / (math.pow(math.sin(k), 2.0) * math.pow(t_m, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	tmp = 0.0
	if (k <= 1.25e-154)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	elseif (k <= 5e+24)
		tmp = Float64(sqrt(2.0) / Float64(Float64((t_m ^ 1.5) / l) * Float64(t_2 * sqrt(Float64(sin(k) * tan(k)))))) ^ 2.0;
	else
		tmp = Float64(Float64(l / t_2) * Float64(2.0 * Float64(Float64(cos(k) * Float64(l / k)) / Float64((sin(k) ^ 2.0) * (t_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	tmp = 0.0;
	if (k <= 1.25e-154)
		tmp = 2.0 / (((((t_m ^ 1.5) * (1.0 / l)) * sqrt(sin(k))) ^ 2.0) * (2.0 * k));
	elseif (k <= 5e+24)
		tmp = (sqrt(2.0) / (((t_m ^ 1.5) / l) * (t_2 * sqrt((sin(k) * tan(k)))))) ^ 2.0;
	else
		tmp = (l / t_2) * (2.0 * ((cos(k) * (l / k)) / ((sin(k) ^ 2.0) * (t_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.25e-154], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+24], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.25000000000000005e-154

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. associate-/r* 34.4%

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

      4. *-commutative 34.4%

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

      5. sqrt-prod 13.3%

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

      6. associate-/r* 11.4%

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

      7. sqrt-div 11.4%

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

      8. sqrt-pow1 15.2%

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

      9. metadata-eval 15.2%

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

      10. sqrt-prod 8.8%

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

      11. add-sqr-sqrt 19.5%

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

   6. Applied egg-rr 19.5%

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

      1. *-commutative 19.5%

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

   8. Simplified 19.5%

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

   9. Taylor expanded in k around 0 15.9%

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

       1. div-inv 15.9%

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

   11. Applied egg-rr 15.9%

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

   if 1.25000000000000005e-154 < k < 5.00000000000000045e24

   1. Initial program 61.8%

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

   3. Simplified 61.8%

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

   6. Applied egg-rr 37.0%

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

   if 5.00000000000000045e24 < k

   1. Initial program 55.0%

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

   3. Simplified 55.2%

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

      1. associate-*r* 61.4%

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

      2. add-sqr-sqrt 61.4%

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

      3. times-frac 61.6%

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

   6. Applied egg-rr 65.9%

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

      1. associate-/l* 73.2%

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

      2. *-commutative 73.2%

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

   8. Simplified 73.2%

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

   9. Taylor expanded in k around inf 67.3%

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

       1. times-frac 69.1%

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

       2. associate-*r/ 69.1%

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

   11. Simplified 69.1%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+24}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-172}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(t\_2 \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t\_m}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

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 (hypot 1.0 (hypot 1.0 (/ k t_m)))))
   (*
    t_s
    (if (<= k 1.6e-172)
      (/
       2.0
       (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
      (if (<= k 4e+28)
        (/
         2.0
         (pow (* (/ (pow t_m 1.5) l) (* t_2 (sqrt (* (sin k) (tan k))))) 2.0))
        (*
         (/ l t_2)
         (*
          2.0
          (/ (* (cos k) (/ l k)) (* (pow (sin k) 2.0) (pow t_m 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (k <= 1.6e-172) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 4e+28) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (t_2 * sqrt((sin(k) * tan(k))))), 2.0);
	} else {
		tmp = (l / t_2) * (2.0 * ((cos(k) * (l / k)) / (pow(sin(k), 2.0) * pow(t_m, 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.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (k <= 1.6e-172) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 4e+28) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (t_2 * Math.sqrt((Math.sin(k) * Math.tan(k))))), 2.0);
	} else {
		tmp = (l / t_2) * (2.0 * ((Math.cos(k) * (l / k)) / (Math.pow(Math.sin(k), 2.0) * Math.pow(t_m, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.hypot(1.0, math.hypot(1.0, (k / t_m)))
	tmp = 0
	if k <= 1.6e-172:
		tmp = 2.0 / (math.pow(((math.pow(t_m, 1.5) * (1.0 / l)) * math.sqrt(math.sin(k))), 2.0) * (2.0 * k))
	elif k <= 4e+28:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (t_2 * math.sqrt((math.sin(k) * math.tan(k))))), 2.0)
	else:
		tmp = (l / t_2) * (2.0 * ((math.cos(k) * (l / k)) / (math.pow(math.sin(k), 2.0) * math.pow(t_m, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	tmp = 0.0
	if (k <= 1.6e-172)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	elseif (k <= 4e+28)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(t_2 * sqrt(Float64(sin(k) * tan(k))))) ^ 2.0));
	else
		tmp = Float64(Float64(l / t_2) * Float64(2.0 * Float64(Float64(cos(k) * Float64(l / k)) / Float64((sin(k) ^ 2.0) * (t_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	tmp = 0.0;
	if (k <= 1.6e-172)
		tmp = 2.0 / (((((t_m ^ 1.5) * (1.0 / l)) * sqrt(sin(k))) ^ 2.0) * (2.0 * k));
	elseif (k <= 4e+28)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (t_2 * sqrt((sin(k) * tan(k))))) ^ 2.0);
	else
		tmp = (l / t_2) * (2.0 * ((cos(k) * (l / k)) / ((sin(k) ^ 2.0) * (t_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.6e-172], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+28], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6000000000000001e-172

   1. Initial program 56.4%

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

   3. Simplified 56.4%

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

      1. add-sqr-sqrt 31.8%

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

      2. pow2 31.8%

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

      3. associate-/r* 34.7%

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

      4. *-commutative 34.7%

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

      5. sqrt-prod 12.6%

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

      6. associate-/r* 10.6%

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

      7. sqrt-div 10.6%

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

      8. sqrt-pow1 14.6%

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

      9. metadata-eval 14.6%

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

      10. sqrt-prod 9.2%

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

      11. add-sqr-sqrt 19.1%

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

   6. Applied egg-rr 19.1%

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

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in k around 0 15.4%

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

       1. div-inv 15.4%

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

   11. Applied egg-rr 15.4%

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

   if 1.6000000000000001e-172 < k < 3.99999999999999983e28

   1. Initial program 65.6%

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

   3. Simplified 65.5%

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

      1. associate-*l* 65.5%

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

      2. associate-/r* 69.4%

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

      3. associate-+r+ 69.4%

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

      4. metadata-eval 69.4%

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

      5. associate-*l* 69.4%

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

      6. add-sqr-sqrt 29.9%

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

      7. pow2 29.9%

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

   6. Applied egg-rr 37.0%

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

   if 3.99999999999999983e28 < k

   1. Initial program 54.1%

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

   3. Simplified 54.3%

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

      1. associate-*r* 60.6%

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

      2. add-sqr-sqrt 60.6%

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

      3. times-frac 60.8%

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

   6. Applied egg-rr 65.2%

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

      1. associate-/l* 72.7%

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

      2. *-commutative 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in k around inf 66.7%

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

       1. times-frac 68.5%

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

       2. associate-*r/ 68.5%

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

   11. Simplified 68.5%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-172}:\\ \;\;\;\;\frac{2}{{\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 16000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-154)
    (/
     2.0
     (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
    (if (<= k 16000000000.0)
      (/
       2.0
       (pow
        (* (/ t_m (cbrt l)) (* (cbrt (/ 1.0 l)) (cbrt (* 2.0 (pow k 2.0)))))
        3.0))
      (*
       (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))
       (* 2.0 (/ (* (cos k) (/ l k)) (* (pow (sin k) 2.0) (pow t_m 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-154) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 16000000000.0) {
		tmp = 2.0 / pow(((t_m / cbrt(l)) * (cbrt((1.0 / l)) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = (l / hypot(1.0, hypot(1.0, (k / t_m)))) * (2.0 * ((cos(k) * (l / k)) / (pow(sin(k), 2.0) * pow(t_m, 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-154) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 16000000000.0) {
		tmp = 2.0 / Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt((1.0 / l)) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = (l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)))) * (2.0 * ((Math.cos(k) * (l / k)) / (Math.pow(Math.sin(k), 2.0) * Math.pow(t_m, 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-154)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	elseif (k <= 16000000000.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(Float64(1.0 / l)) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m)))) * Float64(2.0 * Float64(Float64(cos(k) * Float64(l / k)) / Float64((sin(k) ^ 2.0) * (t_m ^ 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-154], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 16000000000.0], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.20000000000000007e-154

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. associate-/r* 34.4%

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

      4. *-commutative 34.4%

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

      5. sqrt-prod 13.3%

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

      6. associate-/r* 11.4%

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

      7. sqrt-div 11.4%

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

      8. sqrt-pow1 15.2%

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

      9. metadata-eval 15.2%

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

      10. sqrt-prod 8.8%

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

      11. add-sqr-sqrt 19.5%

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

   6. Applied egg-rr 19.5%

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

      1. *-commutative 19.5%

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

   8. Simplified 19.5%

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

   9. Taylor expanded in k around 0 15.9%

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

       1. div-inv 15.9%

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

   11. Applied egg-rr 15.9%

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

   if 2.20000000000000007e-154 < k < 1.6e10

   1. Initial program 61.9%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in k around 0 74.7%

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

      1. associate-*l/ 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. add-cube-cbrt 79.1%

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

      2. pow3 79.0%

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

      3. associate-/l* 78.8%

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

      4. cbrt-prod 78.9%

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

      5. cbrt-div 81.1%

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

      6. unpow3 81.1%

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

      7. add-cbrt-cube 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. pow1/3 42.8%

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

       2. div-inv 42.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\color{blue}{\left(\left(2 \cdot {k}^{2}\right) \cdot \frac{1}{\ell}\right)}}^{0.3333333333333333}\right)}^{3}} \]

       3. unpow-prod-down 44.9%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \color{blue}{\left({\left(2 \cdot {k}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

       4. pow1/3 44.9%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\color{blue}{\sqrt[3]{2 \cdot {k}^{2}}} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   11. Applied egg-rr 44.9%

       \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \color{blue}{\left(\sqrt[3]{2 \cdot {k}^{2}} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   12. Step-by-step derivation

       1. unpow1/3 87.7%

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

       2. *-commutative 87.7%

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

   13. Simplified 87.7%

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

   if 1.6e10 < k

   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 55.8%

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

      1. associate-*r* 61.4%

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

      2. add-sqr-sqrt 61.3%

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

      3. times-frac 61.5%

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

   6. Applied egg-rr 65.4%

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

      1. associate-/l* 73.8%

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

      2. *-commutative 73.8%

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

   8. Simplified 73.8%

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

   9. Taylor expanded in k around inf 66.8%

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

       1. times-frac 68.3%

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

       2. associate-*r/ 68.5%

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

   11. Simplified 68.5%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 16000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(2 \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot {t}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t\_m}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\sin k} \cdot \frac{{\left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\right)}^{2}}{\tan k \cdot {t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-75)
    (/
     2.0
     (* (pow k 2.0) (* (/ t_m (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
    (if (<= t_m 5.6e+76)
      (*
       (/ 2.0 (sin k))
       (/
        (pow (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))) 2.0)
        (* (tan k) (pow t_m 3.0))))
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
        (+ 2.0 (pow (/ k t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-75) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else if (t_m <= 5.6e+76) {
		tmp = (2.0 / sin(k)) * (pow((l / hypot(1.0, hypot(1.0, (k / t_m)))), 2.0) / (tan(k) * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (2.0 + pow((k / t_m), 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 (t_m <= 5.5e-75) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else if (t_m <= 5.6e+76) {
		tmp = (2.0 / Math.sin(k)) * (Math.pow((l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)))), 2.0) / (Math.tan(k) * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (2.0 + Math.pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.5e-75:
		tmp = 2.0 / (math.pow(k, 2.0) * ((t_m / math.pow(l, 2.0)) * (math.pow(math.sin(k), 2.0) / math.cos(k))))
	elif t_m <= 5.6e+76:
		tmp = (2.0 / math.sin(k)) * (math.pow((l / math.hypot(1.0, math.hypot(1.0, (k / t_m)))), 2.0) / (math.tan(k) * math.pow(t_m, 3.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (2.0 + math.pow((k / t_m), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.5e-75)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	elseif (t_m <= 5.6e+76)
		tmp = Float64(Float64(2.0 / sin(k)) * Float64((Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m)))) ^ 2.0) / Float64(tan(k) * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.5e-75)
		tmp = 2.0 / ((k ^ 2.0) * ((t_m / (l ^ 2.0)) * ((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 5.6e+76)
		tmp = (2.0 / sin(k)) * (((l / hypot(1.0, hypot(1.0, (k / t_m)))) ^ 2.0) / (tan(k) * (t_m ^ 3.0)));
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (2.0 + ((k / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-75], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+76], N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.50000000000000026e-75

   1. Initial program 56.1%

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

   3. Simplified 56.5%

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

      1. associate-/r* 51.9%

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

      2. unpow3 51.9%

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

      3. times-frac 60.0%

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

      4. pow2 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 64.2%

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

      2. times-frac 65.9%

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

   9. Simplified 65.9%

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

   if 5.50000000000000026e-75 < t < 5.5999999999999997e76

   1. Initial program 66.2%

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

   3. Simplified 66.1%

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

      1. associate-*r* 74.0%

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

      2. add-sqr-sqrt 73.9%

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

      3. times-frac 74.0%

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

   6. Applied egg-rr 79.5%

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

      1. associate-/l* 86.9%

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

      2. *-commutative 86.9%

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

   8. Simplified 86.9%

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

      1. add-cube-cbrt 86.2%

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

      2. pow3 86.3%

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

      3. *-commutative 86.3%

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

      4. cbrt-prod 86.1%

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

      5. *-commutative 86.1%

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

      6. cbrt-prod 85.9%

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

      7. unpow3 85.9%

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

      8. add-cbrt-cube 85.9%

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

   10. Applied egg-rr 85.9%

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

       1. add-cube-cbrt 85.8%

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

       2. pow3 85.8%

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

   12. Applied egg-rr 85.8%

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

       1. rem-cube-cbrt 85.9%

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

       2. associate-*l* 85.6%

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

   14. Applied egg-rr 89.1%

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

       1. associate-/l* 89.2%

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

       2. *-commutative 89.2%

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

   16. Simplified 89.2%

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

   if 5.5999999999999997e76 < t

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. associate-/r* 34.6%

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

      4. *-commutative 34.6%

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

      5. sqrt-prod 34.6%

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

      6. associate-/r* 30.4%

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

      7. sqrt-div 30.4%

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

      8. sqrt-pow1 41.8%

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

      9. metadata-eval 41.8%

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

      10. sqrt-prod 24.6%

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

      11. add-sqr-sqrt 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. *-commutative 50.9%

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

   8. Simplified 50.9%

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

      1. div-inv 50.9%

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

      2. *-commutative 50.9%

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

      3. unpow-prod-down 50.9%

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

      4. pow2 50.9%

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

      5. add-sqr-sqrt 85.6%

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

      6. pow2 85.6%

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

      7. associate-+r+ 85.6%

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

      8. metadata-eval 85.6%

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

      9. pow2 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. associate-*r/ 85.6%

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

       2. metadata-eval 85.6%

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

       3. associate-*r* 85.6%

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

   12. Simplified 85.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\sin k} \cdot \frac{{\left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}}{\tan k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.1e-155)
    (/
     2.0
     (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
    (if (<= k 2.7e-8)
      (/
       2.0
       (pow
        (* (/ t_m (cbrt l)) (* (cbrt (/ 1.0 l)) (cbrt (* 2.0 (pow k 2.0)))))
        3.0))
      (/
       2.0
       (*
        (pow (/ (pow t_m 1.5) l) 2.0)
        (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.1e-155) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 2.7e-8) {
		tmp = 2.0 / pow(((t_m / cbrt(l)) * (cbrt((1.0 / l)) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.1e-155) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 2.7e-8) {
		tmp = 2.0 / Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt((1.0 / l)) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.1e-155)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	elseif (k <= 2.7e-8)
		tmp = Float64(2.0 / (Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(Float64(1.0 / l)) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.1e-155], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.7e-8], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.10000000000000023e-155

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. associate-/r* 34.4%

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

      4. *-commutative 34.4%

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

      5. sqrt-prod 13.3%

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

      6. associate-/r* 11.4%

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

      7. sqrt-div 11.4%

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

      8. sqrt-pow1 15.2%

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

      9. metadata-eval 15.2%

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

      10. sqrt-prod 8.8%

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

      11. add-sqr-sqrt 19.5%

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

   6. Applied egg-rr 19.5%

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

      1. *-commutative 19.5%

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

   8. Simplified 19.5%

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

   9. Taylor expanded in k around 0 15.9%

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

       1. div-inv 15.9%

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

   11. Applied egg-rr 15.9%

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

   if 6.10000000000000023e-155 < k < 2.70000000000000002e-8

   1. Initial program 61.9%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in k around 0 74.7%

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

      1. associate-*l/ 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. add-cube-cbrt 79.1%

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

      2. pow3 79.0%

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

      3. associate-/l* 78.8%

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

      4. cbrt-prod 78.9%

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

      5. cbrt-div 81.1%

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

      6. unpow3 81.1%

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

      7. add-cbrt-cube 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. pow1/3 42.8%

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

       2. div-inv 42.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\color{blue}{\left(\left(2 \cdot {k}^{2}\right) \cdot \frac{1}{\ell}\right)}}^{0.3333333333333333}\right)}^{3}} \]

       3. unpow-prod-down 44.9%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \color{blue}{\left({\left(2 \cdot {k}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

       4. pow1/3 44.9%

          \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\color{blue}{\sqrt[3]{2 \cdot {k}^{2}}} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   11. Applied egg-rr 44.9%

       \[\leadsto \frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \color{blue}{\left(\sqrt[3]{2 \cdot {k}^{2}} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   12. Step-by-step derivation

       1. unpow1/3 87.7%

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

       2. *-commutative 87.7%

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

   13. Simplified 87.7%

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

   if 2.70000000000000002e-8 < k

   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 61.1%

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

      1. unpow2 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. add-sqr-sqrt 39.6%

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

      2. pow2 39.6%

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

      3. associate-/r* 36.0%

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

      4. sqrt-div 36.0%

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

      5. sqrt-pow1 37.8%

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

      6. metadata-eval 37.8%

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

      7. sqrt-prod 21.4%

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

      8. add-sqr-sqrt 43.1%

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

   8. Applied egg-rr 43.1%

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

Alternative 9: 43.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot t\_2\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{2 \cdot {k}^{2}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_2}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 t_m 1.5) l)))
   (*
    t_s
    (if (<= k 3.6e-147)
      (/ 2.0 (* (* 2.0 k) (pow (* (sqrt (sin k)) t_2) 2.0)))
      (if (<= k 4.5e-6)
        (/
         2.0
         (pow
          (* (/ t_m (cbrt l)) (/ (cbrt (* 2.0 (pow k 2.0))) (cbrt l)))
          3.0))
        (/
         2.0
         (*
          (pow t_2 2.0)
          (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m)))))))))))

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(t_m, 1.5) / l;
	double tmp;
	if (k <= 3.6e-147) {
		tmp = 2.0 / ((2.0 * k) * pow((sqrt(sin(k)) * t_2), 2.0));
	} else if (k <= 4.5e-6) {
		tmp = 2.0 / pow(((t_m / cbrt(l)) * (cbrt((2.0 * pow(k, 2.0))) / cbrt(l))), 3.0);
	} else {
		tmp = 2.0 / (pow(t_2, 2.0) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	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(t_m, 1.5) / l;
	double tmp;
	if (k <= 3.6e-147) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.sqrt(Math.sin(k)) * t_2), 2.0));
	} else if (k <= 4.5e-6) {
		tmp = 2.0 / Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt((2.0 * Math.pow(k, 2.0))) / Math.cbrt(l))), 3.0);
	} else {
		tmp = 2.0 / (Math.pow(t_2, 2.0) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	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((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k <= 3.6e-147)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(sqrt(sin(k)) * t_2) ^ 2.0)));
	elseif (k <= 4.5e-6)
		tmp = Float64(2.0 / (Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(Float64(2.0 * (k ^ 2.0))) / cbrt(l))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64((t_2 ^ 2.0) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.6e-147], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.5e-6], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.60000000000000012e-147

   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 58.2%

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

      1. add-sqr-sqrt 31.9%

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

      2. pow2 31.9%

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

      3. associate-/r* 34.6%

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

      4. *-commutative 34.6%

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

      5. sqrt-prod 13.8%

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

      6. associate-/r* 11.8%

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

      7. sqrt-div 11.8%

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

      8. sqrt-pow1 15.6%

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

      9. metadata-eval 15.6%

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

      10. sqrt-prod 8.7%

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

      11. add-sqr-sqrt 19.8%

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

   6. Applied egg-rr 19.8%

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

      1. *-commutative 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in k around 0 16.3%

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

   if 3.60000000000000012e-147 < k < 4.50000000000000011e-6

   1. Initial program 60.0%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in k around 0 73.4%

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

      1. associate-*l/ 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. add-cube-cbrt 78.0%

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

      2. pow3 78.0%

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

      3. associate-/l* 77.8%

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

      4. cbrt-prod 77.8%

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

      5. cbrt-div 80.2%

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

      6. unpow3 80.2%

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

      7. add-cbrt-cube 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. cbrt-div 87.1%

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

   11. Applied egg-rr 87.1%

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

   if 4.50000000000000011e-6 < k

   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 61.1%

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

      1. unpow2 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. add-sqr-sqrt 39.6%

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

      2. pow2 39.6%

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

      3. associate-/r* 36.0%

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

      4. sqrt-div 36.0%

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

      5. sqrt-pow1 37.8%

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

      6. metadata-eval 37.8%

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

      7. sqrt-prod 21.4%

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

      8. add-sqr-sqrt 43.1%

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

   8. Applied egg-rr 43.1%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{2 \cdot {k}^{2}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t\_m}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-80)
    (/
     2.0
     (* (pow k 2.0) (* (/ t_m (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
    (/
     2.0
     (*
      (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
      (+ 2.0 (pow (/ k t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-80) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (2.0 + pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-80) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((t_m / (l ** 2.0d0)) * ((sin(k) ** 2.0d0) / cos(k))))
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (2.0d0 + ((k / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-80) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (2.0 + Math.pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-80:
		tmp = 2.0 / (math.pow(k, 2.0) * ((t_m / math.pow(l, 2.0)) * (math.pow(math.sin(k), 2.0) / math.cos(k))))
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (2.0 + math.pow((k / t_m), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-80)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-80)
		tmp = 2.0 / ((k ^ 2.0) * ((t_m / (l ^ 2.0)) * ((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (2.0 + ((k / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-80], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000003e-80

   1. Initial program 56.1%

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

   3. Simplified 56.5%

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

      1. associate-/r* 51.9%

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

      2. unpow3 51.9%

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

      3. times-frac 60.0%

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

      4. pow2 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 64.2%

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

      2. times-frac 65.9%

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

   9. Simplified 65.9%

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

   if 4.20000000000000003e-80 < t

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

      1. add-sqr-sqrt 35.4%

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

      2. pow2 35.4%

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

      3. associate-/r* 38.9%

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

      4. *-commutative 38.9%

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

      5. sqrt-prod 38.9%

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

      6. associate-/r* 35.4%

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

      7. sqrt-div 36.4%

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

      8. sqrt-pow1 43.0%

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

      9. metadata-eval 43.0%

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

      10. sqrt-prod 24.1%

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

      11. add-sqr-sqrt 51.5%

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

   6. Applied egg-rr 51.5%

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

      1. *-commutative 51.5%

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

   8. Simplified 51.5%

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

      1. div-inv 51.5%

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

      2. *-commutative 51.5%

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

      3. unpow-prod-down 48.4%

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

      4. pow2 48.4%

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

      5. add-sqr-sqrt 81.7%

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

      6. pow2 81.7%

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

      7. associate-+r+ 81.7%

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

      8. metadata-eval 81.7%

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

      9. pow2 81.7%

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

   10. Applied egg-rr 81.7%

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

       1. associate-*r/ 81.7%

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

       2. metadata-eval 81.7%

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

       3. associate-*r* 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.3 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right)}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot {\left(\ell \cdot \frac{t\_m}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t\_m \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 5.7e-8)
    (/
     2.0
     (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
    (if (<= k 4.3e+94)
      (/
       2.0
       (*
        (* (/ (pow t_m 2.0) l) (/ t_m l))
        (* (* (sin k) (tan k)) (+ 2.0 (/ (* k k) (* t_m t_m))))))
      (if (<= k 1.5e+206)
        (*
         (/ (/ 2.0 (sin k)) (* (tan k) (pow t_m 3.0)))
         (pow (* l (/ t_m k)) 2.0))
        (* l (/ 2.0 (/ (pow (* t_m (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 (k <= 5.7e-8) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 4.3e+94) {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else if (k <= 1.5e+206) {
		tmp = ((2.0 / sin(k)) / (tan(k) * pow(t_m, 3.0))) * pow((l * (t_m / k)), 2.0);
	} else {
		tmp = l * (2.0 / (pow((t_m * 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 (k <= 5.7e-8) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else if (k <= 4.3e+94) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else if (k <= 1.5e+206) {
		tmp = ((2.0 / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * Math.pow((l * (t_m / k)), 2.0);
	} else {
		tmp = l * (2.0 / (Math.pow((t_m * 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 (k <= 5.7e-8)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	elseif (k <= 4.3e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m))))));
	elseif (k <= 1.5e+206)
		tmp = Float64(Float64(Float64(2.0 / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * (Float64(l * Float64(t_m / k)) ^ 2.0));
	else
		tmp = Float64(l * Float64(2.0 / Float64((Float64(t_m * 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[k, 5.7e-8], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.3e+94], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+206], N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 5.70000000000000009e-8

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. associate-/r* 33.4%

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

      4. *-commutative 33.4%

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

      5. sqrt-prod 16.7%

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

      6. associate-/r* 15.1%

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

      7. sqrt-div 15.5%

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

      8. sqrt-pow1 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-prod 12.0%

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

      11. add-sqr-sqrt 22.5%

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

   6. Applied egg-rr 22.5%

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

      1. *-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in k around 0 20.2%

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

       1. div-inv 20.2%

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

   11. Applied egg-rr 20.2%

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

   if 5.70000000000000009e-8 < k < 4.3e94

   1. Initial program 76.4%

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

   3. Simplified 77.0%

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

      1. associate-/r* 76.4%

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

      2. unpow3 76.4%

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

      3. times-frac 82.8%

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

      4. pow2 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. unpow2 82.8%

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

      2. frac-2neg 82.8%

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

      3. frac-times 82.8%

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

   8. Applied egg-rr 82.8%

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

   if 4.3e94 < k < 1.5000000000000001e206

   1. Initial program 50.3%

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

   3. Simplified 50.3%

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

      1. associate-*r* 59.3%

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

      2. add-sqr-sqrt 59.3%

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

      3. times-frac 59.8%

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

   6. Applied egg-rr 69.9%

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

      1. associate-/l* 86.6%

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

      2. *-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. pow1 86.6%

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

      2. associate-*l* 79.4%

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

      3. associate-*l* 79.4%

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

      4. pow2 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. unpow1 79.4%

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

       2. associate-/r* 79.5%

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

   12. Simplified 79.5%

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

   13. Taylor expanded in k around inf 70.5%

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

       1. associate-/l* 74.8%

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

   15. Simplified 74.8%

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

   if 1.5000000000000001e206 < k

   1. Initial program 41.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in k around 0 47.7%

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

      1. associate-*l/ 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. associate-/r/ 47.7%

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

      2. associate-*l/ 47.9%

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

   9. Applied egg-rr 47.9%

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

       1. add-cube-cbrt 47.9%

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

       2. pow3 47.9%

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

       3. cbrt-prod 47.9%

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

       4. unpow3 47.9%

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

       5. add-cbrt-cube 60.8%

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

   11. Applied egg-rr 60.8%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.3 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\tan k \cdot {t}^{3}} \cdot {\left(\ell \cdot \frac{t}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right)}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot {\left(\ell \cdot \frac{t\_m}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t\_m \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 1.3e-7)
    (/ 2.0 (* (* 2.0 k) (pow (* (sqrt (sin k)) (/ (pow t_m 1.5) l)) 2.0)))
    (if (<= k 5.8e+94)
      (/
       2.0
       (*
        (* (/ (pow t_m 2.0) l) (/ t_m l))
        (* (* (sin k) (tan k)) (+ 2.0 (/ (* k k) (* t_m t_m))))))
      (if (<= k 1.7e+204)
        (*
         (/ (/ 2.0 (sin k)) (* (tan k) (pow t_m 3.0)))
         (pow (* l (/ t_m k)) 2.0))
        (* l (/ 2.0 (/ (pow (* t_m (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 (k <= 1.3e-7) {
		tmp = 2.0 / ((2.0 * k) * pow((sqrt(sin(k)) * (pow(t_m, 1.5) / l)), 2.0));
	} else if (k <= 5.8e+94) {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else if (k <= 1.7e+204) {
		tmp = ((2.0 / sin(k)) / (tan(k) * pow(t_m, 3.0))) * pow((l * (t_m / k)), 2.0);
	} else {
		tmp = l * (2.0 / (pow((t_m * 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 (k <= 1.3e-7) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.sqrt(Math.sin(k)) * (Math.pow(t_m, 1.5) / l)), 2.0));
	} else if (k <= 5.8e+94) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else if (k <= 1.7e+204) {
		tmp = ((2.0 / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * Math.pow((l * (t_m / k)), 2.0);
	} else {
		tmp = l * (2.0 / (Math.pow((t_m * 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 (k <= 1.3e-7)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(sqrt(sin(k)) * Float64((t_m ^ 1.5) / l)) ^ 2.0)));
	elseif (k <= 5.8e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m))))));
	elseif (k <= 1.7e+204)
		tmp = Float64(Float64(Float64(2.0 / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * (Float64(l * Float64(t_m / k)) ^ 2.0));
	else
		tmp = Float64(l * Float64(2.0 / Float64((Float64(t_m * 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[k, 1.3e-7], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e+94], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e+204], N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.29999999999999999e-7

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. associate-/r* 33.4%

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

      4. *-commutative 33.4%

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

      5. sqrt-prod 16.7%

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

      6. associate-/r* 15.1%

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

      7. sqrt-div 15.5%

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

      8. sqrt-pow1 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-prod 12.0%

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

      11. add-sqr-sqrt 22.5%

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

   6. Applied egg-rr 22.5%

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

      1. *-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in k around 0 20.2%

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

   if 1.29999999999999999e-7 < k < 5.7999999999999997e94

   1. Initial program 76.4%

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

   3. Simplified 77.0%

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

      1. associate-/r* 76.4%

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

      2. unpow3 76.4%

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

      3. times-frac 82.8%

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

      4. pow2 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. unpow2 82.8%

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

      2. frac-2neg 82.8%

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

      3. frac-times 82.8%

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

   8. Applied egg-rr 82.8%

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

   if 5.7999999999999997e94 < k < 1.70000000000000005e204

   1. Initial program 50.3%

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

   3. Simplified 50.3%

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

      1. associate-*r* 59.3%

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

      2. add-sqr-sqrt 59.3%

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

      3. times-frac 59.8%

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

   6. Applied egg-rr 69.9%

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

      1. associate-/l* 86.6%

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

      2. *-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. pow1 86.6%

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

      2. associate-*l* 79.4%

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

      3. associate-*l* 79.4%

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

      4. pow2 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. unpow1 79.4%

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

       2. associate-/r* 79.5%

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

   12. Simplified 79.5%

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

   13. Taylor expanded in k around inf 70.5%

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

       1. associate-/l* 74.8%

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

   15. Simplified 74.8%

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

   if 1.70000000000000005e204 < k

   1. Initial program 41.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in k around 0 47.7%

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

      1. associate-*l/ 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. associate-/r/ 47.7%

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

      2. associate-*l/ 47.9%

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

   9. Applied egg-rr 47.9%

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

       1. add-cube-cbrt 47.9%

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

       2. pow3 47.9%

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

       3. cbrt-prod 47.9%

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

       4. unpow3 47.9%

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

       5. add-cbrt-cube 60.8%

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

   11. Applied egg-rr 60.8%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\tan k \cdot {t}^{3}} \cdot {\left(\ell \cdot \frac{t}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.8e-5)
    (/
     2.0
     (* (pow (* (* (pow t_m 1.5) (/ 1.0 l)) (sqrt (sin k))) 2.0) (* 2.0 k)))
    (/
     2.0
     (*
      (pow (/ (pow t_m 1.5) l) 2.0)
      (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m)))))))))

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 <= 4.8e-5) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) * (1.0 / l)) * sqrt(sin(k))), 2.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	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 <= 4.8d-5) then
        tmp = 2.0d0 / (((((t_m ** 1.5d0) * (1.0d0 / l)) * sqrt(sin(k))) ** 2.0d0) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * ((sin(k) * tan(k)) * (2.0d0 + ((k / t_m) * (k / t_m)))))
    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 <= 4.8e-5) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) * (1.0 / l)) * Math.sqrt(Math.sin(k))), 2.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	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 <= 4.8e-5:
		tmp = 2.0 / (math.pow(((math.pow(t_m, 1.5) * (1.0 / l)) * math.sqrt(math.sin(k))), 2.0) * (2.0 * k))
	else:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))))
	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 <= 4.8e-5)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) * Float64(1.0 / l)) * sqrt(sin(k))) ^ 2.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	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 <= 4.8e-5)
		tmp = 2.0 / (((((t_m ^ 1.5) * (1.0 / l)) * sqrt(sin(k))) ^ 2.0) * (2.0 * k));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	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, 4.8e-5], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.8000000000000001e-5

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. associate-/r* 33.4%

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

      4. *-commutative 33.4%

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

      5. sqrt-prod 16.7%

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

      6. associate-/r* 15.1%

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

      7. sqrt-div 15.5%

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

      8. sqrt-pow1 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-prod 12.0%

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

      11. add-sqr-sqrt 22.5%

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

   6. Applied egg-rr 22.5%

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

      1. *-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in k around 0 20.2%

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

       1. div-inv 20.2%

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

   11. Applied egg-rr 20.2%

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

   if 4.8000000000000001e-5 < k

   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 61.1%

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

      1. unpow2 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. add-sqr-sqrt 39.6%

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

      2. pow2 39.6%

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

      3. associate-/r* 36.0%

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

      4. sqrt-div 36.0%

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

      5. sqrt-pow1 37.8%

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

      6. metadata-eval 37.8%

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

      7. sqrt-prod 21.4%

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

      8. add-sqr-sqrt 43.1%

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

   8. Applied egg-rr 43.1%

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

Alternative 14: 41.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-8)
    (/ 2.0 (* (* 2.0 k) (pow (* (sqrt (sin k)) (/ (pow t_m 1.5) l)) 2.0)))
    (/
     2.0
     (*
      (* (/ (pow t_m 2.0) l) (/ t_m l))
      (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-8) {
		tmp = 2.0 / ((2.0 * k) * pow((sqrt(sin(k)) * (pow(t_m, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-8) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sqrt(sin(k)) * ((t_m ** 1.5d0) / l)) ** 2.0d0))
    else
        tmp = 2.0d0 / ((((t_m ** 2.0d0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0d0 + ((k / t_m) / (t_m / k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-8) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.sqrt(Math.sin(k)) * (Math.pow(t_m, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-8:
		tmp = 2.0 / ((2.0 * k) * math.pow((math.sqrt(math.sin(k)) * (math.pow(t_m, 1.5) / l)), 2.0))
	else:
		tmp = 2.0 / (((math.pow(t_m, 2.0) / l) * (t_m / l)) * ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-8)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(sqrt(sin(k)) * Float64((t_m ^ 1.5) / l)) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-8)
		tmp = 2.0 / ((2.0 * k) * ((sqrt(sin(k)) * ((t_m ^ 1.5) / l)) ^ 2.0));
	else
		tmp = 2.0 / ((((t_m ^ 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-8], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999986e-8

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. associate-/r* 33.4%

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

      4. *-commutative 33.4%

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

      5. sqrt-prod 16.7%

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

      6. associate-/r* 15.1%

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

      7. sqrt-div 15.5%

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

      8. sqrt-pow1 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-prod 12.0%

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

      11. add-sqr-sqrt 22.5%

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

   6. Applied egg-rr 22.5%

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

      1. *-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in k around 0 20.2%

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

   if 8.99999999999999986e-8 < k

   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 61.1%

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

      1. associate-/r* 55.5%

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

      2. unpow3 55.5%

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

      3. times-frac 64.6%

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

      4. pow2 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow2 64.6%

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

      2. clear-num 64.6%

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

      3. un-div-inv 64.5%

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

   8. Applied egg-rr 64.5%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t\_m \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left({k}^{2} \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-162)
    (* l (/ 2.0 (pow (* t_m (cbrt (* 2.0 (/ (pow k 2.0) l)))) 3.0)))
    (if (<= t_m 8e-113)
      (/
       2.0
       (*
        (* (/ (pow t_m 2.0) l) (/ t_m l))
        (* (pow k 2.0) (+ 2.0 (pow (/ k t_m) 2.0)))))
      (if (<= t_m 7.1e+75)
        (/
         2.0
         (*
          (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m))))
          (/ (/ (pow t_m 3.0) l) l)))
        (/ 2.0 (* (sin k) (* (* 2.0 k) (pow (/ (pow t_m 1.5) l) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-162) {
		tmp = l * (2.0 / pow((t_m * cbrt((2.0 * (pow(k, 2.0) / l)))), 3.0));
	} else if (t_m <= 8e-113) {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * (pow(k, 2.0) * (2.0 + pow((k / t_m), 2.0))));
	} else if (t_m <= 7.1e+75) {
		tmp = 2.0 / (((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))) * ((pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / (sin(k) * ((2.0 * k) * pow((pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-162) {
		tmp = l * (2.0 / Math.pow((t_m * Math.cbrt((2.0 * (Math.pow(k, 2.0) / l)))), 3.0));
	} else if (t_m <= 8e-113) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * (Math.pow(k, 2.0) * (2.0 + Math.pow((k / t_m), 2.0))));
	} else if (t_m <= 7.1e+75) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))) * ((Math.pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-162)
		tmp = Float64(l * Float64(2.0 / (Float64(t_m * cbrt(Float64(2.0 * Float64((k ^ 2.0) / l)))) ^ 3.0)));
	elseif (t_m <= 8e-113)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64((k ^ 2.0) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (t_m <= 7.1e+75)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-162], N[(l * N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e-113], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.1e+75], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.30000000000000013e-162

   1. Initial program 56.4%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in k around 0 58.7%

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

      1. associate-*l/ 60.0%

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

   7. Applied egg-rr 60.0%

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

      1. associate-/r/ 60.0%

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

      2. associate-*l/ 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. add-cube-cbrt 59.9%

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

       2. pow3 60.0%

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

       3. associate-/l* 59.8%

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

       4. cbrt-prod 59.8%

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

       5. unpow3 59.8%

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

       6. add-cbrt-cube 67.2%

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

       7. associate-/l* 67.2%

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

   11. Applied egg-rr 67.2%

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

   if 3.30000000000000013e-162 < t < 7.99999999999999983e-113

   1. Initial program 50.0%

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

   3. Simplified 60.2%

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

      1. associate-/r* 50.0%

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

      2. unpow3 50.0%

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

      3. times-frac 70.8%

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

      4. pow2 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in k around 0 70.8%

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

   if 7.99999999999999983e-113 < t < 7.09999999999999982e75

   1. Initial program 64.2%

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

   3. Simplified 72.8%

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

      1. unpow2 72.8%

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

   6. Applied egg-rr 72.8%

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

   if 7.09999999999999982e75 < t

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. associate-/r* 35.7%

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

      4. *-commutative 35.7%

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

      5. sqrt-prod 35.7%

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

      6. associate-/r* 31.7%

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

      7. sqrt-div 31.7%

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

      8. sqrt-pow1 42.8%

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

      9. metadata-eval 42.8%

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

      10. sqrt-prod 24.1%

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

      11. add-sqr-sqrt 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. *-commutative 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in k around 0 46.6%

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

       1. div-inv 46.6%

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

       2. *-commutative 46.6%

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

       3. unpow-prod-down 46.6%

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

       4. pow2 46.6%

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

       5. add-sqr-sqrt 80.7%

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

   11. Applied egg-rr 80.7%

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

       1. associate-*r/ 80.7%

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

       2. metadata-eval 80.7%

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

       3. associate-*l* 80.7%

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

   13. Simplified 80.7%

       \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot k\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t\_m \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-162)
    (* l (/ 2.0 (pow (* t_m (cbrt (* 2.0 (/ (pow k 2.0) l)))) 3.0)))
    (if (<= t_m 6.5e+146)
      (/
       2.0
       (*
        (* (/ (pow t_m 2.0) l) (/ t_m l))
        (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))))
      (/ 2.0 (* (sin k) (* (* 2.0 k) (pow (/ (pow t_m 1.5) l) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-162) {
		tmp = l * (2.0 / pow((t_m * cbrt((2.0 * (pow(k, 2.0) / l)))), 3.0));
	} else if (t_m <= 6.5e+146) {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / (sin(k) * ((2.0 * k) * pow((pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-162) {
		tmp = l * (2.0 / Math.pow((t_m * Math.cbrt((2.0 * (Math.pow(k, 2.0) / l)))), 3.0));
	} else if (t_m <= 6.5e+146) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-162)
		tmp = Float64(l * Float64(2.0 / (Float64(t_m * cbrt(Float64(2.0 * Float64((k ^ 2.0) / l)))) ^ 3.0)));
	elseif (t_m <= 6.5e+146)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-162], N[(l * N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+146], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.30000000000000013e-162

   1. Initial program 56.4%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in k around 0 58.7%

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

      1. associate-*l/ 60.0%

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

   7. Applied egg-rr 60.0%

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

      1. associate-/r/ 60.0%

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

      2. associate-*l/ 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. add-cube-cbrt 59.9%

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

       2. pow3 60.0%

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

       3. associate-/l* 59.8%

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

       4. cbrt-prod 59.8%

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

       5. unpow3 59.8%

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

       6. add-cbrt-cube 67.2%

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

       7. associate-/l* 67.2%

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

   11. Applied egg-rr 67.2%

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

   if 3.30000000000000013e-162 < t < 6.4999999999999997e146

   1. Initial program 59.5%

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

   3. Simplified 66.0%

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

      1. associate-/r* 59.2%

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

      2. unpow3 59.3%

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

      3. times-frac 76.9%

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

      4. pow2 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. unpow2 76.9%

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

      2. clear-num 77.0%

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

      3. un-div-inv 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 6.4999999999999997e146 < t

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

      1. add-sqr-sqrt 37.1%

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

      2. pow2 37.1%

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

      3. associate-/r* 43.2%

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

      4. *-commutative 43.2%

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

      5. sqrt-prod 43.2%

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

      6. associate-/r* 37.1%

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

      7. sqrt-div 37.1%

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

      8. sqrt-pow1 40.4%

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

      9. metadata-eval 40.4%

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

      10. sqrt-prod 23.0%

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

      11. add-sqr-sqrt 51.5%

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

   6. Applied egg-rr 51.5%

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

      1. *-commutative 51.5%

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

   8. Simplified 51.5%

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

   9. Taylor expanded in k around 0 48.8%

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

       1. div-inv 48.8%

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

       2. *-commutative 48.8%

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

       3. unpow-prod-down 48.8%

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

       4. pow2 48.8%

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

       5. add-sqr-sqrt 78.3%

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

   11. Applied egg-rr 78.3%

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

       1. associate-*r/ 78.3%

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

       2. metadata-eval 78.3%

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

       3. associate-*l* 78.3%

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

   13. Simplified 78.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t\_m \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e-131)
    (/ 2.0 (* k (* (* 2.0 k) (pow (/ (pow t_m 1.5) l) 2.0))))
    (* l (/ 2.0 (pow (* t_m (cbrt (* 2.0 (/ (pow k 2.0) l)))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e-131) {
		tmp = 2.0 / (k * ((2.0 * k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = l * (2.0 / pow((t_m * cbrt((2.0 * (pow(k, 2.0) / l)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e-131) {
		tmp = 2.0 / (k * ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = l * (2.0 / Math.pow((t_m * Math.cbrt((2.0 * (Math.pow(k, 2.0) / l)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.2e-131)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(l * Float64(2.0 / (Float64(t_m * cbrt(Float64(2.0 * Float64((k ^ 2.0) / l)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.2e-131], N[(2.0 / N[(k * N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000087e-131

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

      1. add-sqr-sqrt 31.5%

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

      2. pow2 31.5%

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

      3. associate-/r* 34.2%

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

      4. *-commutative 34.2%

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

      5. sqrt-prod 13.6%

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

      6. associate-/r* 11.7%

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

      7. sqrt-div 11.7%

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

      8. sqrt-pow1 15.4%

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

      9. metadata-eval 15.4%

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

      10. sqrt-prod 8.6%

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

      11. add-sqr-sqrt 19.6%

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

   6. Applied egg-rr 19.6%

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

      1. *-commutative 19.6%

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

   8. Simplified 19.6%

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

   9. Taylor expanded in k around 0 16.1%

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

       1. div-inv 16.1%

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

       2. *-commutative 16.1%

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

       3. unpow-prod-down 15.0%

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

       4. pow2 15.0%

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

       5. add-sqr-sqrt 32.6%

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

   11. Applied egg-rr 32.6%

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

       1. associate-*r/ 32.6%

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

       2. metadata-eval 32.6%

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

       3. associate-*l* 32.6%

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

   13. Simplified 32.6%

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

   14. Taylor expanded in k around 0 35.5%

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

   if 9.20000000000000087e-131 < k

   1. Initial program 56.4%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in k around 0 56.5%

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

      1. associate-*l/ 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. associate-/r/ 58.5%

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

      2. associate-*l/ 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. add-cube-cbrt 58.5%

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

       2. pow3 58.5%

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

       3. associate-/l* 59.5%

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

       4. cbrt-prod 59.5%

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

       5. unpow3 59.5%

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

       6. add-cbrt-cube 63.0%

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

       7. associate-/l* 62.9%

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

   11. Applied egg-rr 62.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t\_m \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 2.75e-105)
    (/ 2.0 (* k (* (* 2.0 k) (pow (/ (pow t_m 1.5) l) 2.0))))
    (* l (/ 2.0 (/ (pow (* t_m (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 (k <= 2.75e-105) {
		tmp = 2.0 / (k * ((2.0 * k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = l * (2.0 / (pow((t_m * 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 (k <= 2.75e-105) {
		tmp = 2.0 / (k * ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = l * (2.0 / (Math.pow((t_m * 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 (k <= 2.75e-105)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(l * Float64(2.0 / Float64((Float64(t_m * 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[k, 2.75e-105], N[(2.0 / N[(k * N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.75000000000000015e-105

   1. Initial program 59.6%

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

   3. Simplified 59.6%

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

      1. add-sqr-sqrt 32.1%

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

      2. pow2 32.1%

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

      3. associate-/r* 34.8%

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

      4. *-commutative 34.8%

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

      5. sqrt-prod 14.9%

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

      6. associate-/r* 13.0%

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

      7. sqrt-div 13.0%

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

      8. sqrt-pow1 16.7%

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

      9. metadata-eval 16.7%

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

      10. sqrt-prod 8.9%

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

      11. add-sqr-sqrt 20.7%

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

   6. Applied egg-rr 20.7%

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

      1. *-commutative 20.7%

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

   8. Simplified 20.7%

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

   9. Taylor expanded in k around 0 17.3%

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

       1. div-inv 17.3%

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

       2. *-commutative 17.3%

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

       3. unpow-prod-down 16.2%

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

       4. pow2 16.2%

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

       5. add-sqr-sqrt 33.2%

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

   11. Applied egg-rr 33.2%

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

       1. associate-*r/ 33.2%

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

       2. metadata-eval 33.2%

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

       3. associate-*l* 33.2%

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

   13. Simplified 33.2%

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

   14. Taylor expanded in k around 0 36.0%

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

   if 2.75000000000000015e-105 < k

   1. Initial program 54.5%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in k around 0 53.6%

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

      1. associate-*l/ 55.7%

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

   7. Applied egg-rr 55.7%

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

      1. associate-/r/ 55.7%

         \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell} \]

      2. associate-*l/ 56.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \cdot \ell \]

   9. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}} \cdot \ell} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 56.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \sqrt[3]{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\right) \cdot \sqrt[3]{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}}}{\ell}} \cdot \ell \]

       2. pow3 56.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\right)}^{3}}}{\ell}} \cdot \ell \]

       3. cbrt-prod 56.7%

          \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}}^{3}}{\ell}} \cdot \ell \]

       4. unpow3 56.7%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}} \cdot \ell \]

       5. add-cbrt-cube 60.4%

          \[\leadsto \frac{2}{\frac{{\left(\color{blue}{t} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}} \cdot \ell \]

   11. Applied egg-rr 60.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{{\left(t \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}}{\ell}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{\left(t \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)} \end{array} \]

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 (* k (* (* 2.0 k) (pow (/ (pow t_m 1.5) l) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (k * ((2.0 * k) * pow((pow(t_m, 1.5) / l), 2.0))));
}

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 / (k * ((2.0d0 * k) * (((t_m ** 1.5d0) / l) ** 2.0d0))))
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 / (k * ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))));
}

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 / (k * ((2.0 * k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(k * Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)))))
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 / (k * ((2.0 * k) * (((t_m ^ 1.5) / l) ^ 2.0))));
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[(2.0 / N[(k * N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 57.9%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   2. pow2 32.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   3. associate-/r* 34.4%

      \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. *-commutative 34.4%

      \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. sqrt-prod 18.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\sin k} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. associate-/r* 16.9%

      \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. sqrt-div 17.3%

      \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. sqrt-pow1 20.4%

      \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   9. metadata-eval 20.4%

      \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   10. sqrt-prod 11.7%

       \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   11. add-sqr-sqrt 23.8%

       \[\leadsto \frac{2}{{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

6. Applied egg-rr 23.8%

   \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 23.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}}^{2} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

8. Simplified 23.8%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

9. Taylor expanded in k around 0 19.5%

   \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
10. Step-by-step derivation

    1. div-inv 19.5%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \left(2 \cdot k\right)}} \]

    2. *-commutative 19.5%

       \[\leadsto 2 \cdot \frac{1}{{\color{blue}{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}}^{2} \cdot \left(2 \cdot k\right)} \]

    3. unpow-prod-down 18.4%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left({\left(\sqrt{\sin k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \left(2 \cdot k\right)} \]

    4. pow2 18.4%

       \[\leadsto 2 \cdot \frac{1}{\left(\color{blue}{\left(\sqrt{\sin k} \cdot \sqrt{\sin k}\right)} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)} \]

    5. add-sqr-sqrt 32.0%

       \[\leadsto 2 \cdot \frac{1}{\left(\color{blue}{\sin k} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)} \]

11. Applied egg-rr 32.0%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}} \]
12. Step-by-step derivation

    1. associate-*r/ 32.0%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}} \]

    2. metadata-eval 32.0%

       \[\leadsto \frac{\color{blue}{2}}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)} \]

    3. associate-*l* 32.0%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot k\right)\right)}} \]

13. Simplified 32.0%

    \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot k\right)\right)}} \]

14. Taylor expanded in k around 0 34.2%

    \[\leadsto \frac{2}{\color{blue}{k} \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot k\right)\right)} \]

15. Final simplification 34.2%

    \[\leadsto \frac{2}{k \cdot \left(\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]
16. Add Preprocessing

Alternative 20: 55.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot {t\_m}^{3}}\right) \end{array} \]

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 (* l (/ l (* (pow k 2.0) (pow t_m 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (pow(k, 2.0) * pow(t_m, 3.0))));
}

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 * (l * (l / ((k ** 2.0d0) * (t_m ** 3.0d0))))
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 * (l * (l / (Math.pow(k, 2.0) * Math.pow(t_m, 3.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (math.pow(k, 2.0) * math.pow(t_m, 3.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64((k ^ 2.0) * (t_m ^ 3.0)))))
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 * (l * (l / ((k ^ 2.0) * (t_m ^ 3.0))));
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[(l * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 57.8%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*l/ 58.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

7. Applied egg-rr 58.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
8. Step-by-step derivation

   1. associate-/r/ 58.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell} \]

   2. associate-*l/ 58.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \cdot \ell \]

9. Applied egg-rr 58.2%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}} \cdot \ell} \]

10. Taylor expanded in t around 0 58.3%

    \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]

11. Final simplification 58.3%

    \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{3}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(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))))