Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 87.0%

Time: 21.7s

Alternatives: 19

Speedup: 3.8×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\\ t_3 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t\_m}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{1}{\frac{\ell}{{t\_m}^{2}}}\right)\right) \cdot t\_2}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;t\_3 \cdot \left(\frac{\frac{2}{\tan k}}{\sin k \cdot {t\_m}^{3}} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
        (t_3 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 3.6e-151)
      (/ 2.0 (/ (* (pow k 2.0) (/ (* (/ t_m l) (pow (sin k) 2.0)) (cos k))) l))
      (if (<= t_m 5.4e-76)
        (/ 2.0 (* (* (sin k) (* (/ t_m l) (/ 1.0 (/ l (pow t_m 2.0))))) t_2))
        (if (<= t_m 1.4e+99)
          (* t_3 (* (/ (/ 2.0 (tan k)) (* (sin k) (pow t_m 3.0))) t_3))
          (/
           2.0
           (*
            t_2
            (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)));
	double t_3 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = 2.0 / ((pow(k, 2.0) * (((t_m / l) * pow(sin(k), 2.0)) / cos(k))) / l);
	} else if (t_m <= 5.4e-76) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (1.0 / (l / pow(t_m, 2.0))))) * t_2);
	} else if (t_m <= 1.4e+99) {
		tmp = t_3 * (((2.0 / tan(k)) / (sin(k) * pow(t_m, 3.0))) * t_3);
	} else {
		tmp = 2.0 / (t_2 * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)));
	double t_3 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (((t_m / l) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	} else if (t_m <= 5.4e-76) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (1.0 / (l / Math.pow(t_m, 2.0))))) * t_2);
	} else if (t_m <= 1.4e+99) {
		tmp = t_3 * (((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow(t_m, 3.0))) * t_3);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))))
	t_3 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 3.6e-151)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(Float64(t_m / l) * (sin(k) ^ 2.0)) / cos(k))) / l));
	elseif (t_m <= 5.4e-76)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64(1.0 / Float64(l / (t_m ^ 2.0))))) * t_2));
	elseif (t_m <= 1.4e+99)
		tmp = Float64(t_3 * Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (t_m ^ 3.0))) * t_3));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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, 3.6e-151], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.4e-76], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+99], N[(t$95$3 * N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.60000000000000032e-151

   1. Initial program 57.4%

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

   3. Simplified 57.4%

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

      1. associate-*l* 51.0%

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

      2. associate-/r* 56.1%

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

      3. associate-+r+ 56.1%

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

      4. metadata-eval 56.1%

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

      5. associate-*l* 56.1%

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

      6. associate-*l/ 59.1%

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

      7. associate-*r* 59.0%

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

   6. Applied egg-rr 59.0%

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

   7. Taylor expanded in t around 0 68.2%

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

      1. associate-/l* 68.2%

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

      2. times-frac 71.0%

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

      3. associate-*r/ 71.0%

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

   9. Simplified 71.0%

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

   if 3.60000000000000032e-151 < t < 5.4000000000000001e-76

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. unpow3 47.1%

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

      2. times-frac 80.6%

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

      3. pow2 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. clear-num 80.7%

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

      2. inv-pow 80.7%

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

   8. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

   10. Simplified 80.7%

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

   if 5.4000000000000001e-76 < t < 1.4e99

   1. Initial program 73.8%

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

   3. Simplified 76.6%

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

      1. associate-*r* 79.4%

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

      2. add-sqr-sqrt 79.4%

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

      3. times-frac 85.4%

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

   6. Applied egg-rr 96.6%

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

      1. associate-/l* 96.7%

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

      2. metadata-eval 96.7%

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

      3. distribute-neg-frac 96.7%

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

      4. associate-/l/ 96.8%

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

      5. distribute-neg-frac 96.8%

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

      6. distribute-neg-frac 96.8%

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

      7. metadata-eval 96.8%

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

   8. Simplified 96.8%

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

   if 1.4e99 < t

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

      1. add-cube-cbrt 53.9%

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

      2. pow3 53.9%

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

      3. *-commutative 53.9%

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

      4. cbrt-prod 53.9%

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

      5. cbrt-div 53.9%

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

      6. rem-cbrt-cube 68.5%

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

      7. cbrt-prod 93.8%

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

      8. pow2 93.8%

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

   6. Applied egg-rr 93.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{{t}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t\_m}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{1}{\frac{\ell}{{t\_m}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;t\_3 \cdot \left(t\_3 \cdot \frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + t\_2\right)}\right)\right)}^{3}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 3.6e-151)
      (/ 2.0 (/ (* (pow k 2.0) (/ (* (/ t_m l) (pow (sin k) 2.0)) (cos k))) l))
      (if (<= t_m 2e-81)
        (/
         2.0
         (*
          (* (sin k) (* (/ t_m l) (/ 1.0 (/ l (pow t_m 2.0)))))
          (* (tan k) (+ 1.0 (+ 1.0 t_2)))))
        (if (<= t_m 1.6e+24)
          (* t_3 (* t_3 (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k))))))
          (/
           2.0
           (/
            (pow
             (*
              (/ t_m (cbrt l))
              (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 t_2)))))
             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 t_2 = pow((k / t_m), 2.0);
	double t_3 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = 2.0 / ((pow(k, 2.0) * (((t_m / l) * pow(sin(k), 2.0)) / cos(k))) / l);
	} else if (t_m <= 2e-81) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (1.0 / (l / pow(t_m, 2.0))))) * (tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.6e+24) {
		tmp = t_3 * (t_3 * (2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))));
	} else {
		tmp = 2.0 / (pow(((t_m / cbrt(l)) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + t_2))))), 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 t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (((t_m / l) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	} else if (t_m <= 2e-81) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (1.0 / (l / Math.pow(t_m, 2.0))))) * (Math.tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.6e+24) {
		tmp = t_3 * (t_3 * (2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + t_2))))), 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)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 3.6e-151)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(Float64(t_m / l) * (sin(k) ^ 2.0)) / cos(k))) / l));
	elseif (t_m <= 2e-81)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64(1.0 / Float64(l / (t_m ^ 2.0))))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	elseif (t_m <= 1.6e+24)
		tmp = Float64(t_3 * Float64(t_3 * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + t_2))))) ^ 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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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, 3.6e-151], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e-81], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+24], N[(t$95$3 * N[(t$95$3 * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.60000000000000032e-151

   1. Initial program 57.4%

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

   3. Simplified 57.4%

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

      1. associate-*l* 51.0%

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

      2. associate-/r* 56.1%

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

      3. associate-+r+ 56.1%

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

      4. metadata-eval 56.1%

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

      5. associate-*l* 56.1%

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

      6. associate-*l/ 59.1%

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

      7. associate-*r* 59.0%

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

   6. Applied egg-rr 59.0%

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

   7. Taylor expanded in t around 0 68.2%

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

      1. associate-/l* 68.2%

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

      2. times-frac 71.0%

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

      3. associate-*r/ 71.0%

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

   9. Simplified 71.0%

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

   if 3.60000000000000032e-151 < t < 1.9999999999999999e-81

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. unpow3 47.1%

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

      2. times-frac 80.6%

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

      3. pow2 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. clear-num 80.7%

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

      2. inv-pow 80.7%

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

   8. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

   10. Simplified 80.7%

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

   if 1.9999999999999999e-81 < t < 1.5999999999999999e24

   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 66.2%

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

      1. associate-*r* 66.4%

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

      2. add-sqr-sqrt 66.4%

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

      3. times-frac 76.2%

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

   6. Applied egg-rr 94.8%

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

      1. associate-/l* 95.0%

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

      2. associate-*l* 95.0%

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

   8. Simplified 95.0%

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

   if 1.5999999999999999e24 < t

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

      1. associate-*l* 55.9%

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

      2. associate-/r* 60.9%

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

      3. associate-+r+ 60.9%

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

      4. metadata-eval 60.9%

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

      5. associate-*l* 60.9%

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

      6. associate-*l/ 62.8%

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

      7. associate-*r* 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. add-cube-cbrt 62.7%

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

      2. pow3 62.7%

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

      3. associate-*l* 62.7%

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

      4. cbrt-prod 62.6%

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

      5. cbrt-div 62.7%

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

      6. unpow3 62.7%

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

      7. add-cbrt-cube 71.3%

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

      8. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. metadata-eval 71.3%

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

      3. associate-+r+ 71.3%

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

      4. cbrt-prod 89.4%

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

      5. associate-+r+ 89.4%

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

      6. metadata-eval 89.4%

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

   10. Applied egg-rr 89.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{{t}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 26.5:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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 2e-147)
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0)))
    (if (<= k 26.5)
      (/
       2.0
       (pow
        (*
         (/ (pow t_m 1.5) l)
         (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (sqrt (* (sin k) (tan k)))))
        2.0))
      (/
       2.0
       (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) 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 <= 2e-147) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0));
	} else if (k <= 26.5) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k / t_m))) * sqrt((sin(k) * tan(k))))), 2.0);
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / 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 <= 2e-147) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0));
	} else if (k <= 26.5) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * Math.sqrt((Math.sin(k) * Math.tan(k))))), 2.0);
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / 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 <= 2e-147)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0)));
	elseif (k <= 26.5)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * sqrt(Float64(sin(k) * tan(k))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / 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, 2e-147], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 26.5], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999999e-147

   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 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.4%

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

      5. cbrt-div 59.9%

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

      6. rem-cbrt-cube 67.8%

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

      7. cbrt-prod 80.3%

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

      8. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in k around 0 78.9%

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

   if 1.9999999999999999e-147 < k < 26.5

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   6. Applied egg-rr 32.1%

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

   if 26.5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 56.3%

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

      3. associate-+r+ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*l* 56.3%

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

      6. associate-*l/ 57.9%

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

      7. associate-*r* 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 83.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 26.5:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t\_m}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{1}{\frac{\ell}{{t\_m}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + t\_2} \cdot \left(\ell \cdot {\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t\_m \cdot \sqrt[3]{\sin k}}\right)}^{3}\right)\\ \end{array} \end{array} \end{array} \]

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 (pow (/ k t_m) 2.0))
        (t_3
         (/
          2.0
          (/ (* (pow k 2.0) (/ (* (/ t_m l) (pow (sin k) 2.0)) (cos k))) l))))
   (*
    t_s
    (if (<= t_m 3.5e-151)
      t_3
      (if (<= t_m 9e-75)
        (/
         2.0
         (*
          (* (sin k) (* (/ t_m l) (/ 1.0 (/ l (pow t_m 2.0)))))
          (* (tan k) (+ 1.0 (+ 1.0 t_2)))))
        (if (<= t_m 1.2e-18)
          t_3
          (*
           (/ l (+ 2.0 t_2))
           (*
            l
            (pow (/ (cbrt (/ 2.0 (tan k))) (* t_m (cbrt (sin 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 = pow((k / t_m), 2.0);
	double t_3 = 2.0 / ((pow(k, 2.0) * (((t_m / l) * pow(sin(k), 2.0)) / cos(k))) / l);
	double tmp;
	if (t_m <= 3.5e-151) {
		tmp = t_3;
	} else if (t_m <= 9e-75) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (1.0 / (l / pow(t_m, 2.0))))) * (tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.2e-18) {
		tmp = t_3;
	} else {
		tmp = (l / (2.0 + t_2)) * (l * pow((cbrt((2.0 / tan(k))) / (t_m * cbrt(sin(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 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 / ((Math.pow(k, 2.0) * (((t_m / l) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	double tmp;
	if (t_m <= 3.5e-151) {
		tmp = t_3;
	} else if (t_m <= 9e-75) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (1.0 / (l / Math.pow(t_m, 2.0))))) * (Math.tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.2e-18) {
		tmp = t_3;
	} else {
		tmp = (l / (2.0 + t_2)) * (l * Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t_m * Math.cbrt(Math.sin(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(k / t_m) ^ 2.0
	t_3 = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(Float64(t_m / l) * (sin(k) ^ 2.0)) / cos(k))) / l))
	tmp = 0.0
	if (t_m <= 3.5e-151)
		tmp = t_3;
	elseif (t_m <= 9e-75)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64(1.0 / Float64(l / (t_m ^ 2.0))))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	elseif (t_m <= 1.2e-18)
		tmp = t_3;
	else
		tmp = Float64(Float64(l / Float64(2.0 + t_2)) * Float64(l * (Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t_m * cbrt(sin(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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-151], t$95$3, If[LessEqual[t$95$m, 9e-75], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-18], t$95$3, N[(N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l * N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.49999999999999995e-151 or 9.0000000000000006e-75 < t < 1.19999999999999997e-18

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

      1. associate-*l* 52.2%

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

      2. associate-/r* 56.9%

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

      3. associate-+r+ 56.9%

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

      4. metadata-eval 56.9%

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

      5. associate-*l* 56.9%

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

      6. associate-*l/ 60.1%

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

      7. associate-*r* 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in t around 0 69.6%

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

      1. associate-/l* 69.6%

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

      2. times-frac 72.2%

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

      3. associate-*r/ 72.2%

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

   9. Simplified 72.2%

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

   if 3.49999999999999995e-151 < t < 9.0000000000000006e-75

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. unpow3 47.1%

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

      2. times-frac 80.6%

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

      3. pow2 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. clear-num 80.7%

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

      2. inv-pow 80.7%

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

   8. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

   10. Simplified 80.7%

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

   if 1.19999999999999997e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 65.0%

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

      1. associate-*r* 71.2%

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

      2. *-un-lft-identity 71.2%

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

      3. times-frac 73.1%

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

      4. associate-/l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. /-rgt-identity 73.1%

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

      2. *-commutative 73.1%

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

      3. metadata-eval 73.1%

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

      4. distribute-neg-frac 73.1%

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

      5. associate-/l/ 73.1%

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

      6. distribute-neg-frac 73.1%

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

      7. distribute-neg-frac 73.1%

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

      8. metadata-eval 73.1%

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

   8. Simplified 73.1%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.0%

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

      3. cbrt-div 72.9%

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

      4. cbrt-prod 72.8%

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

      5. rem-cbrt-cube 83.4%

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

   10. Applied egg-rr 83.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{{t}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot {\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t\_m}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{1}{\frac{\ell}{{t\_m}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\tan k}}}{t\_m \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \frac{\ell}{2 + t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3
         (/
          2.0
          (/ (* (pow k 2.0) (/ (* (/ t_m l) (pow (sin k) 2.0)) (cos k))) l))))
   (*
    t_s
    (if (<= t_m 3.6e-151)
      t_3
      (if (<= t_m 1.25e-79)
        (/
         2.0
         (*
          (* (sin k) (* (/ t_m l) (/ 1.0 (/ l (pow t_m 2.0)))))
          (* (tan k) (+ 1.0 (+ 1.0 t_2)))))
        (if (<= t_m 1.3e-18)
          t_3
          (*
           (pow (/ (cbrt (* l (/ 2.0 (tan k)))) (* t_m (cbrt (sin k)))) 3.0)
           (/ l (+ 2.0 t_2)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = 2.0 / ((pow(k, 2.0) * (((t_m / l) * pow(sin(k), 2.0)) / cos(k))) / l);
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = t_3;
	} else if (t_m <= 1.25e-79) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (1.0 / (l / pow(t_m, 2.0))))) * (tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.3e-18) {
		tmp = t_3;
	} else {
		tmp = pow((cbrt((l * (2.0 / tan(k)))) / (t_m * cbrt(sin(k)))), 3.0) * (l / (2.0 + 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 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 / ((Math.pow(k, 2.0) * (((t_m / l) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	double tmp;
	if (t_m <= 3.6e-151) {
		tmp = t_3;
	} else if (t_m <= 1.25e-79) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (1.0 / (l / Math.pow(t_m, 2.0))))) * (Math.tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 1.3e-18) {
		tmp = t_3;
	} else {
		tmp = Math.pow((Math.cbrt((l * (2.0 / Math.tan(k)))) / (t_m * Math.cbrt(Math.sin(k)))), 3.0) * (l / (2.0 + t_2));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(Float64(t_m / l) * (sin(k) ^ 2.0)) / cos(k))) / l))
	tmp = 0.0
	if (t_m <= 3.6e-151)
		tmp = t_3;
	elseif (t_m <= 1.25e-79)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64(1.0 / Float64(l / (t_m ^ 2.0))))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	elseif (t_m <= 1.3e-18)
		tmp = t_3;
	else
		tmp = Float64((Float64(cbrt(Float64(l * Float64(2.0 / tan(k)))) / Float64(t_m * cbrt(sin(k)))) ^ 3.0) * Float64(l / Float64(2.0 + 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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.6e-151], t$95$3, If[LessEqual[t$95$m, 1.25e-79], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e-18], t$95$3, N[(N[Power[N[(N[Power[N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.60000000000000032e-151 or 1.25e-79 < t < 1.3e-18

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

      1. associate-*l* 52.2%

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

      2. associate-/r* 56.9%

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

      3. associate-+r+ 56.9%

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

      4. metadata-eval 56.9%

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

      5. associate-*l* 56.9%

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

      6. associate-*l/ 60.1%

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

      7. associate-*r* 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in t around 0 69.6%

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

      1. associate-/l* 69.6%

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

      2. times-frac 72.2%

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

      3. associate-*r/ 72.2%

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

   9. Simplified 72.2%

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

   if 3.60000000000000032e-151 < t < 1.25e-79

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. unpow3 47.1%

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

      2. times-frac 80.6%

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

      3. pow2 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. clear-num 80.7%

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

      2. inv-pow 80.7%

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

   8. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

   10. Simplified 80.7%

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

   if 1.3e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 65.0%

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

      1. associate-*r* 71.2%

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

      2. *-un-lft-identity 71.2%

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

      3. times-frac 73.1%

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

      4. associate-/l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. /-rgt-identity 73.1%

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

      2. *-commutative 73.1%

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

      3. metadata-eval 73.1%

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

      4. distribute-neg-frac 73.1%

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

      5. associate-/l/ 73.1%

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

      6. distribute-neg-frac 73.1%

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

      7. distribute-neg-frac 73.1%

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

      8. metadata-eval 73.1%

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

   8. Simplified 73.1%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.0%

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

      3. associate-*r/ 72.6%

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

      4. cbrt-div 72.6%

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

      5. cbrt-prod 72.5%

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

      6. unpow3 72.5%

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

      7. add-cbrt-cube 83.4%

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

   10. Applied egg-rr 83.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{{t}^{2}}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\tan k}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t\_m}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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 4e-143)
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0)))
    (if (<= k 3.5e-5)
      (/
       2.0
       (/
        (*
         t_m
         (+ (* 2.0 (/ (* (pow k 2.0) (pow t_m 2.0)) l)) (/ (pow k 4.0) l)))
        l))
      (/
       2.0
       (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) 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 <= 4e-143) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0));
	} else if (k <= 3.5e-5) {
		tmp = 2.0 / ((t_m * ((2.0 * ((pow(k, 2.0) * pow(t_m, 2.0)) / l)) + (pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / 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 <= 4e-143) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0));
	} else if (k <= 3.5e-5) {
		tmp = 2.0 / ((t_m * ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 2.0)) / l)) + (Math.pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / 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 <= 4e-143)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0)));
	elseif (k <= 3.5e-5)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 2.0)) / l)) + Float64((k ^ 4.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / 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, 4e-143], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5e-5], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.9999999999999998e-143

   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 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.4%

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

      5. cbrt-div 59.9%

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

      6. rem-cbrt-cube 67.8%

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

      7. cbrt-prod 80.3%

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

      8. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in k around 0 78.9%

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

   if 3.9999999999999998e-143 < k < 3.4999999999999997e-5

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

      1. associate-*l* 64.8%

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

      2. associate-/r* 65.1%

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

      3. associate-+r+ 65.1%

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

      4. metadata-eval 65.1%

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

      5. associate-*l* 65.1%

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

      6. associate-*l/ 69.0%

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

      7. associate-*r* 69.0%

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

   6. Applied egg-rr 69.0%

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

   7. Taylor expanded in k around 0 69.0%

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

   8. Taylor expanded in t around 0 95.7%

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

   if 3.4999999999999997e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 56.3%

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

      3. associate-+r+ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*l* 56.3%

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

      6. associate-*l/ 57.9%

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

      7. associate-*r* 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 83.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \sqrt[3]{k}\right)\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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 4e-5)
    (/
     2.0
     (/
      (pow
       (*
        (/ t_m (cbrt l))
        (* (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt k)))
       3.0)
      l))
    (/
     2.0
     (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) 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 <= 4e-5) {
		tmp = 2.0 / (pow(((t_m / cbrt(l)) * (cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))) * cbrt(k))), 3.0) / l);
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / 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 <= 4e-5) {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.cbrt(k))), 3.0) / l);
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / 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 <= 4e-5)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * cbrt(k))) ^ 3.0) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / 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, 4e-5], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000033e-5

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. associate-*l* 53.2%

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

      2. associate-/r* 58.5%

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

      3. associate-+r+ 58.5%

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

      4. metadata-eval 58.5%

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

      5. associate-*l* 58.5%

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

      6. associate-*l/ 61.9%

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

      7. associate-*r* 61.9%

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

   6. Applied egg-rr 61.9%

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

      1. add-cube-cbrt 61.8%

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

      2. pow3 61.8%

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

      3. associate-*l* 61.8%

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

      4. cbrt-prod 61.7%

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

      5. cbrt-div 61.8%

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

      6. unpow3 61.8%

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

      7. add-cbrt-cube 71.9%

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

      8. associate-*l* 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. *-commutative 71.9%

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

      2. metadata-eval 71.9%

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

      3. associate-+r+ 71.9%

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

      4. cbrt-prod 82.5%

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

      5. associate-+r+ 82.5%

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

      6. metadata-eval 82.5%

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

   10. Applied egg-rr 82.5%

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

   11. Taylor expanded in k around 0 79.6%

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

   if 4.00000000000000033e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 56.3%

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

      3. associate-+r+ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*l* 56.3%

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

      6. associate-*l/ 57.9%

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

      7. associate-*r* 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 83.0%

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

4. Final simplification 80.4%

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

Alternative 8: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t\_m}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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 4e-143)
    (/ l (pow (* (/ t_m (cbrt l)) (pow (cbrt k) 2.0)) 3.0))
    (if (<= k 3.2e-7)
      (/
       2.0
       (/
        (*
         t_m
         (+ (* 2.0 (/ (* (pow k 2.0) (pow t_m 2.0)) l)) (/ (pow k 4.0) l)))
        l))
      (/
       2.0
       (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) 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 <= 4e-143) {
		tmp = l / pow(((t_m / cbrt(l)) * pow(cbrt(k), 2.0)), 3.0);
	} else if (k <= 3.2e-7) {
		tmp = 2.0 / ((t_m * ((2.0 * ((pow(k, 2.0) * pow(t_m, 2.0)) / l)) + (pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / 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 <= 4e-143) {
		tmp = l / Math.pow(((t_m / Math.cbrt(l)) * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	} else if (k <= 3.2e-7) {
		tmp = 2.0 / ((t_m * ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 2.0)) / l)) + (Math.pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / 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 <= 4e-143)
		tmp = Float64(l / (Float64(Float64(t_m / cbrt(l)) * (cbrt(k) ^ 2.0)) ^ 3.0));
	elseif (k <= 3.2e-7)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 2.0)) / l)) + Float64((k ^ 4.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / 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, 4e-143], N[(l / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.2e-7], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.9999999999999998e-143

   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 59.5%

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

      1. associate-*l* 51.4%

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

      2. associate-/r* 57.6%

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

      3. associate-+r+ 57.6%

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

      4. metadata-eval 57.6%

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

      5. associate-*l* 57.6%

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

      6. associate-*l/ 60.8%

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

      7. associate-*r* 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in k around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-/l* 57.2%

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

   9. Simplified 57.2%

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

       1. *-un-lft-identity 57.2%

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

       2. associate-/r/ 56.7%

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

       3. associate-*r* 56.7%

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

   11. Applied egg-rr 56.7%

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

       1. *-lft-identity 56.7%

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

       2. associate-*l/ 57.2%

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

       3. associate-*l* 57.2%

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

       4. times-frac 57.2%

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

       5. metadata-eval 57.2%

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

       6. associate-*r/ 57.1%

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

       7. *-commutative 57.1%

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

       8. associate-/l* 57.7%

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

   13. Simplified 57.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 57.1%

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

       2. add-cube-cbrt 57.0%

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

       3. pow3 57.0%

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

       4. associate-*r/ 57.7%

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

       5. *-commutative 57.7%

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

       6. cbrt-prod 57.6%

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

       7. cbrt-div 57.7%

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

       8. unpow3 57.7%

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

       9. add-cbrt-cube 62.8%

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

       10. unpow2 62.8%

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

       11. cbrt-prod 72.2%

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

       12. pow2 72.2%

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

   15. Applied egg-rr 72.2%

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

   if 3.9999999999999998e-143 < k < 3.2000000000000001e-7

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

      1. associate-*l* 64.8%

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

      2. associate-/r* 65.1%

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

      3. associate-+r+ 65.1%

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

      4. metadata-eval 65.1%

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

      5. associate-*l* 65.1%

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

      6. associate-*l/ 69.0%

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

      7. associate-*r* 69.0%

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

   6. Applied egg-rr 69.0%

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

   7. Taylor expanded in k around 0 69.0%

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

   8. Taylor expanded in t around 0 95.7%

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

   if 3.2000000000000001e-7 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 56.3%

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

      3. associate-+r+ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*l* 56.3%

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

      6. associate-*l/ 57.9%

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

      7. associate-*r* 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 83.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.22e-18)
    (/ 2.0 (/ (* (/ (pow k 2.0) l) (/ (* t_m (pow (sin k) 2.0)) (cos k))) l))
    (/ l (pow (* (/ t_m (cbrt l)) (pow (cbrt k) 2.0)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.22e-18) {
		tmp = 2.0 / (((pow(k, 2.0) / l) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l);
	} else {
		tmp = l / pow(((t_m / cbrt(l)) * pow(cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.22e-18) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	} else {
		tmp = l / Math.pow(((t_m / Math.cbrt(l)) * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.22e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l));
	else
		tmp = Float64(l / (Float64(Float64(t_m / cbrt(l)) * (cbrt(k) ^ 2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.22e-18], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2200000000000001e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in t around 0 71.6%

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

      1. times-frac 70.3%

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

   9. Simplified 70.3%

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

   if 1.2200000000000001e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 59.7%

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

       2. add-cube-cbrt 59.6%

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

       3. pow3 59.6%

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

       4. associate-*r/ 59.6%

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

       5. *-commutative 59.6%

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

       6. cbrt-prod 59.6%

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

       7. cbrt-div 59.7%

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

       8. unpow3 59.6%

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

       9. add-cbrt-cube 65.5%

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

       10. unpow2 65.5%

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

       11. cbrt-prod 81.3%

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

       12. pow2 81.3%

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

   15. Applied egg-rr 81.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t\_m}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-18)
    (/ 2.0 (/ (* (pow k 2.0) (/ (* (/ t_m l) (pow (sin k) 2.0)) (cos k))) l))
    (/ l (pow (* (/ t_m (cbrt l)) (pow (cbrt k) 2.0)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-18) {
		tmp = 2.0 / ((pow(k, 2.0) * (((t_m / l) * pow(sin(k), 2.0)) / cos(k))) / l);
	} else {
		tmp = l / pow(((t_m / cbrt(l)) * pow(cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-18) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (((t_m / l) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	} else {
		tmp = l / Math.pow(((t_m / Math.cbrt(l)) * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-18)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(Float64(t_m / l) * (sin(k) ^ 2.0)) / cos(k))) / l));
	else
		tmp = Float64(l / (Float64(Float64(t_m / cbrt(l)) * (cbrt(k) ^ 2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-18], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999994e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in t around 0 71.6%

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

      1. associate-/l* 71.3%

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

      2. times-frac 73.5%

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

      3. associate-*r/ 73.5%

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

   9. Simplified 73.5%

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

   if 1.34999999999999994e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 59.7%

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

       2. add-cube-cbrt 59.6%

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

       3. pow3 59.6%

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

       4. associate-*r/ 59.6%

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

       5. *-commutative 59.6%

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

       6. cbrt-prod 59.6%

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

       7. cbrt-div 59.7%

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

       8. unpow3 59.6%

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

       9. add-cbrt-cube 65.5%

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

       10. unpow2 65.5%

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

       11. cbrt-prod 81.3%

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

       12. pow2 81.3%

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

   15. Applied egg-rr 81.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.15e-18)
    (/ 2.0 (/ (* (/ t_m l) (pow k 4.0)) l))
    (/ l (pow (* (/ t_m (cbrt l)) (pow (cbrt k) 2.0)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.15e-18) {
		tmp = 2.0 / (((t_m / l) * pow(k, 4.0)) / l);
	} else {
		tmp = l / pow(((t_m / cbrt(l)) * pow(cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.15e-18) {
		tmp = 2.0 / (((t_m / l) * Math.pow(k, 4.0)) / l);
	} else {
		tmp = l / Math.pow(((t_m / Math.cbrt(l)) * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.15e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * (k ^ 4.0)) / l));
	else
		tmp = Float64(l / (Float64(Float64(t_m / cbrt(l)) * (cbrt(k) ^ 2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.15e-18], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in k around 0 53.6%

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

   8. Taylor expanded in k around inf 59.2%

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

      1. associate-/l* 60.5%

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

   10. Simplified 60.5%

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

   if 1.15e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 59.7%

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

       2. add-cube-cbrt 59.6%

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

       3. pow3 59.6%

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

       4. associate-*r/ 59.6%

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

       5. *-commutative 59.6%

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

       6. cbrt-prod 59.6%

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

       7. cbrt-div 59.7%

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

       8. unpow3 59.6%

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

       9. add-cbrt-cube 65.5%

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

       10. unpow2 65.5%

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

       11. cbrt-prod 81.3%

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

       12. pow2 81.3%

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

   15. Applied egg-rr 81.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{t\_m}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t\_m}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {\left(t\_m \cdot \sqrt[3]{\frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 3.5e-146)
    (/ 2.0 (* (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l))) (* 2.0 k)))
    (if (<= k 1.15e+49)
      (/
       2.0
       (/
        (*
         t_m
         (+ (* 2.0 (/ (* (pow k 2.0) (pow t_m 2.0)) l)) (/ (pow k 4.0) l)))
        l))
      (/ 2.0 (/ (* 2.0 (pow (* t_m (cbrt (/ (pow k 2.0) l))) 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 <= 3.5e-146) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))) * (2.0 * k));
	} else if (k <= 1.15e+49) {
		tmp = 2.0 / ((t_m * ((2.0 * ((pow(k, 2.0) * pow(t_m, 2.0)) / l)) + (pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / ((2.0 * pow((t_m * cbrt((pow(k, 2.0) / l))), 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 <= 3.5e-146) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))) * (2.0 * k));
	} else if (k <= 1.15e+49) {
		tmp = 2.0 / ((t_m * ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 2.0)) / l)) + (Math.pow(k, 4.0) / l))) / l);
	} else {
		tmp = 2.0 / ((2.0 * Math.pow((t_m * Math.cbrt((Math.pow(k, 2.0) / l))), 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 <= 3.5e-146)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l))) * Float64(2.0 * k)));
	elseif (k <= 1.15e+49)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 2.0)) / l)) + Float64((k ^ 4.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (Float64(t_m * cbrt(Float64((k ^ 2.0) / l))) ^ 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, 3.5e-146], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+49], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[N[(t$95$m * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.5000000000000001e-146

   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 59.5%

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

      1. unpow3 59.5%

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

      2. times-frac 72.5%

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

      3. pow2 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in k around 0 69.1%

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

   if 3.5000000000000001e-146 < k < 1.15000000000000001e49

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. associate-*l* 61.2%

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

      2. associate-/r* 61.5%

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

      3. associate-+r+ 61.5%

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

      4. metadata-eval 61.5%

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

      5. associate-*l* 61.5%

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

      6. associate-*l/ 64.5%

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

      7. associate-*r* 64.5%

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

   6. Applied egg-rr 64.5%

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

   7. Taylor expanded in k around 0 55.7%

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

   8. Taylor expanded in t around 0 79.1%

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

   if 1.15000000000000001e49 < k

   1. Initial program 51.5%

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

   3. Simplified 51.5%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.2%

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

      3. associate-+r+ 57.2%

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

      4. metadata-eval 57.2%

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

      5. associate-*l* 57.2%

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

      6. associate-*l/ 59.0%

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

      7. associate-*r* 58.9%

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

   6. Applied egg-rr 58.9%

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

   7. Taylor expanded in k around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. associate-/l* 50.4%

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

   9. Simplified 50.4%

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

       1. add-cube-cbrt 50.4%

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

       2. pow3 50.4%

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

       3. cbrt-prod 50.4%

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

       4. unpow3 50.4%

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

       5. add-cbrt-cube 63.5%

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

   11. Applied egg-rr 63.5%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{k}^{4}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {\left(t \cdot \sqrt[3]{\frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{t\_m}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {\left(t\_m \cdot \sqrt[3]{\frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 4e-144)
    (/ 2.0 (* (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l))) (* 2.0 k)))
    (/ 2.0 (/ (* 2.0 (pow (* t_m (cbrt (/ (pow k 2.0) l))) 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 <= 4e-144) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * pow((t_m * cbrt((pow(k, 2.0) / l))), 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 <= 4e-144) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow((t_m * Math.cbrt((Math.pow(k, 2.0) / l))), 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 <= 4e-144)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l))) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (Float64(t_m * cbrt(Float64((k ^ 2.0) / l))) ^ 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, 4e-144], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[N[(t$95$m * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.9999999999999998e-144

   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 59.5%

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

      1. unpow3 59.5%

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

      2. times-frac 72.5%

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

      3. pow2 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in k around 0 69.1%

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

   if 3.9999999999999998e-144 < k

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

      1. associate-*l* 55.1%

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

      2. associate-/r* 58.8%

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

      3. associate-+r+ 58.8%

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

      4. metadata-eval 58.8%

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

      5. associate-*l* 58.8%

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

      6. associate-*l/ 61.0%

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

      7. associate-*r* 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in k around 0 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-/l* 53.5%

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

   9. Simplified 53.5%

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

       1. add-cube-cbrt 53.5%

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

       2. pow3 53.5%

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

       3. cbrt-prod 53.4%

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

       4. unpow3 53.4%

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

       5. add-cbrt-cube 67.0%

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

   11. Applied egg-rr 67.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {\left(t \cdot \sqrt[3]{\frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{t\_m}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m \cdot {k}^{4}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 3.15e+56)
    (/ 2.0 (* (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l))) (* 2.0 k)))
    (/ 2.0 (/ (/ (* t_m (pow k 4.0)) l) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.15e+56) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t_m * pow(k, 4.0)) / l) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.15d+56) then
        tmp = 2.0d0 / ((sin(k) * ((t_m / l) * ((t_m ** 2.0d0) / l))) * (2.0d0 * k))
    else
        tmp = 2.0d0 / (((t_m * (k ** 4.0d0)) / l) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.15e+56) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k, 4.0)) / l) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.15e+56:
		tmp = 2.0 / ((math.sin(k) * ((t_m / l) * (math.pow(t_m, 2.0) / l))) * (2.0 * k))
	else:
		tmp = 2.0 / (((t_m * math.pow(k, 4.0)) / l) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.15e+56)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l))) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 4.0)) / l) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.15e+56)
		tmp = 2.0 / ((sin(k) * ((t_m / l) * ((t_m ^ 2.0) / l))) * (2.0 * k));
	else
		tmp = 2.0 / (((t_m * (k ^ 4.0)) / l) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.15e+56], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.15e56

   1. Initial program 59.9%

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

   3. Simplified 59.9%

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

      1. unpow3 59.9%

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

      2. times-frac 72.1%

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

      3. pow2 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in k around 0 68.4%

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

   if 3.15e56 < k

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

      1. associate-*l* 50.5%

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

      2. associate-/r* 56.5%

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

      3. associate-+r+ 56.5%

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

      4. metadata-eval 56.5%

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

      5. associate-*l* 56.6%

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

      6. associate-*l/ 56.6%

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

      7. associate-*r* 56.6%

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in k around 0 51.2%

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

   8. Taylor expanded in k around inf 65.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-35)
    (/ 2.0 (/ (* (/ t_m l) (pow k 4.0)) l))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-35) {
		tmp = 2.0 / (((t_m / l) * pow(k, 4.0)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.1d-35) then
        tmp = 2.0d0 / (((t_m / l) * (k ** 4.0d0)) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-35) {
		tmp = 2.0 / (((t_m / l) * Math.pow(k, 4.0)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.1e-35:
		tmp = 2.0 / (((t_m / l) * math.pow(k, 4.0)) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-35)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * (k ^ 4.0)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.1e-35)
		tmp = 2.0 / (((t_m / l) * (k ^ 4.0)) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-35], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.09999999999999997e-35

   1. Initial program 56.0%

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

   3. Simplified 56.0%

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

      1. associate-*l* 50.8%

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

      2. associate-/r* 56.4%

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

      3. associate-+r+ 56.4%

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

      4. metadata-eval 56.4%

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

      5. associate-*l* 56.4%

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

      6. associate-*l/ 59.2%

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

      7. associate-*r* 59.2%

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

   6. Applied egg-rr 59.2%

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

   7. Taylor expanded in k around 0 53.4%

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

   8. Taylor expanded in k around inf 59.1%

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

      1. associate-/l* 60.4%

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

   10. Simplified 60.4%

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

   if 1.09999999999999997e-35 < t

   1. Initial program 65.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in k around 0 63.5%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2} \cdot \frac{1}{\frac{\ell}{{t\_m}^{3}}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-18)
    (/ 2.0 (/ (* (/ t_m l) (pow k 4.0)) l))
    (/ l (* (pow k 2.0) (/ 1.0 (/ l (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) {
	double tmp;
	if (t_m <= 1.2e-18) {
		tmp = 2.0 / (((t_m / l) * pow(k, 4.0)) / l);
	} else {
		tmp = l / (pow(k, 2.0) * (1.0 / (l / pow(t_m, 3.0))));
	}
	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 <= 1.2d-18) then
        tmp = 2.0d0 / (((t_m / l) * (k ** 4.0d0)) / l)
    else
        tmp = l / ((k ** 2.0d0) * (1.0d0 / (l / (t_m ** 3.0d0))))
    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 <= 1.2e-18) {
		tmp = 2.0 / (((t_m / l) * Math.pow(k, 4.0)) / l);
	} else {
		tmp = l / (Math.pow(k, 2.0) * (1.0 / (l / Math.pow(t_m, 3.0))));
	}
	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 <= 1.2e-18:
		tmp = 2.0 / (((t_m / l) * math.pow(k, 4.0)) / l)
	else:
		tmp = l / (math.pow(k, 2.0) * (1.0 / (l / math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.2e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * (k ^ 4.0)) / l));
	else
		tmp = Float64(l / Float64((k ^ 2.0) * Float64(1.0 / Float64(l / (t_m ^ 3.0)))));
	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 <= 1.2e-18)
		tmp = 2.0 / (((t_m / l) * (k ^ 4.0)) / l);
	else
		tmp = l / ((k ^ 2.0) * (1.0 / (l / (t_m ^ 3.0))));
	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, 1.2e-18], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(1.0 / N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999997e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in k around 0 53.6%

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

   8. Taylor expanded in k around inf 59.2%

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

      1. associate-/l* 60.5%

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

   10. Simplified 60.5%

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

   if 1.19999999999999997e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. clear-num 59.8%

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

       2. inv-pow 59.8%

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

   15. Applied egg-rr 59.8%

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

       1. unpow-1 59.8%

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

   17. Simplified 59.8%

       \[\leadsto 1 \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{{t}^{3}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2} \cdot \frac{1}{\frac{\ell}{{t}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{k}^{2} \cdot {t\_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-18)
    (/ 2.0 (/ (* (/ t_m l) (pow k 4.0)) l))
    (* 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) {
	double tmp;
	if (t_m <= 1.35e-18) {
		tmp = 2.0 / (((t_m / l) * pow(k, 4.0)) / l);
	} else {
		tmp = l * (l / (pow(k, 2.0) * pow(t_m, 3.0)));
	}
	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 <= 1.35d-18) then
        tmp = 2.0d0 / (((t_m / l) * (k ** 4.0d0)) / l)
    else
        tmp = l * (l / ((k ** 2.0d0) * (t_m ** 3.0d0)))
    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 <= 1.35e-18) {
		tmp = 2.0 / (((t_m / l) * Math.pow(k, 4.0)) / l);
	} else {
		tmp = l * (l / (Math.pow(k, 2.0) * Math.pow(t_m, 3.0)));
	}
	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 <= 1.35e-18:
		tmp = 2.0 / (((t_m / l) * math.pow(k, 4.0)) / l)
	else:
		tmp = l * (l / (math.pow(k, 2.0) * math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * (k ^ 4.0)) / l));
	else
		tmp = Float64(l * Float64(l / Float64((k ^ 2.0) * (t_m ^ 3.0))));
	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 <= 1.35e-18)
		tmp = 2.0 / (((t_m / l) * (k ^ 4.0)) / l);
	else
		tmp = l * (l / ((k ^ 2.0) * (t_m ^ 3.0)));
	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, 1.35e-18], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if t < 1.34999999999999994e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in k around 0 53.6%

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

   8. Taylor expanded in k around inf 59.2%

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

      1. associate-/l* 60.5%

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

   10. Simplified 60.5%

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

   if 1.34999999999999994e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
   14. Step-by-step derivation

       1. div-inv 59.7%

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

   15. Applied egg-rr 59.7%

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

       1. associate-*r/ 59.7%

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

       2. *-commutative 59.7%

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

       3. *-rgt-identity 59.7%

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

       4. associate-/l/ 57.9%

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

       5. associate-/r/ 58.0%

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

       6. associate-/r* 59.7%

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

   17. Simplified 59.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.3e-18)
    (/ 2.0 (/ (* (/ t_m l) (pow k 4.0)) l))
    (/ l (* (pow k 2.0) (/ (pow t_m 3.0) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.3e-18) {
		tmp = 2.0 / (((t_m / l) * pow(k, 4.0)) / l);
	} else {
		tmp = l / (pow(k, 2.0) * (pow(t_m, 3.0) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.3d-18) then
        tmp = 2.0d0 / (((t_m / l) * (k ** 4.0d0)) / l)
    else
        tmp = l / ((k ** 2.0d0) * ((t_m ** 3.0d0) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.3e-18) {
		tmp = 2.0 / (((t_m / l) * Math.pow(k, 4.0)) / l);
	} else {
		tmp = l / (Math.pow(k, 2.0) * (Math.pow(t_m, 3.0) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.3e-18:
		tmp = 2.0 / (((t_m / l) * math.pow(k, 4.0)) / l)
	else:
		tmp = l / (math.pow(k, 2.0) * (math.pow(t_m, 3.0) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.3e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * (k ^ 4.0)) / l));
	else
		tmp = Float64(l / Float64((k ^ 2.0) * Float64((t_m ^ 3.0) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.3e-18)
		tmp = 2.0 / (((t_m / l) * (k ^ 4.0)) / l);
	else
		tmp = l / ((k ^ 2.0) * ((t_m ^ 3.0) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-18], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-18

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

      1. associate-*l* 51.5%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. associate-*l/ 59.8%

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

      7. associate-*r* 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in k around 0 53.6%

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

   8. Taylor expanded in k around inf 59.2%

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

      1. associate-/l* 60.5%

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

   10. Simplified 60.5%

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

   if 1.3e-18 < t

   1. Initial program 63.1%

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

   3. Simplified 63.2%

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

      1. associate-*l* 57.2%

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

      2. associate-/r* 61.6%

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

      3. associate-+r+ 61.6%

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

      4. metadata-eval 61.6%

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

      5. associate-*l* 61.6%

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

      6. associate-*l/ 65.1%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/l* 58.0%

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

   9. Simplified 58.0%

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

       1. *-un-lft-identity 58.0%

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

       2. associate-/r/ 58.0%

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

       3. associate-*r* 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

       2. associate-*l/ 58.0%

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

       3. associate-*l* 58.0%

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

       4. times-frac 58.0%

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

       5. metadata-eval 58.0%

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

       6. associate-*r/ 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 59.7%

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

   13. Simplified 59.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.1% accurate, 3.8× speedup

Math

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

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 (/ (* (/ t_m l) (pow k 4.0)) l))))

C

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

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 / (((t_m / l) * (k ** 4.0d0)) / l))
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 / (((t_m / l) * Math.pow(k, 4.0)) / l));
}

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 / (((t_m / l) * math.pow(k, 4.0)) / l))

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(Float64(Float64(t_m / l) * (k ^ 4.0)) / l)))
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 / (((t_m / l) * (k ^ 4.0)) / l));
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[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

3. Simplified 58.0%

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

   1. associate-*l* 52.7%

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

   2. associate-/r* 58.0%

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

   3. associate-+r+ 58.0%

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

   4. metadata-eval 58.0%

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

   5. associate-*l* 58.0%

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

   6. associate-*l/ 60.9%

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

   7. associate-*r* 60.9%

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

6. Applied egg-rr 60.9%

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

7. Taylor expanded in k around 0 54.9%

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

8. Taylor expanded in k around inf 56.7%

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

   1. associate-/l* 57.6%

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

10. Simplified 57.6%

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

11. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2024076 
(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))))