Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 83.7%

Time: 16.9s

Alternatives: 20

Speedup: 24.8×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\left(\left(t\_m \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\right) \cdot \left(\ell \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (if (<= t_m 2.95e-5)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (if (<= t_m 5.6e+102)
          (*
           (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0)))
           (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
          (*
           (*
            (/ 2.0 (pow (* (* t_m (cbrt (sin k))) (cbrt (tan k))) 3.0))
            (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))))
           (* l (sqrt 0.5)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else if (t_m <= 2.95e-5) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = ((2.0 / pow(((t_m * cbrt(sin(k))) * cbrt(tan(k))), 3.0)) * (l / hypot(1.0, hypot(1.0, (k / t_m))))) * (l * sqrt(0.5));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else if (t_m <= 2.95e-5) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = ((2.0 / Math.pow(((t_m * Math.cbrt(Math.sin(k))) * Math.cbrt(Math.tan(k))), 3.0)) * (l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m))))) * (l * Math.sqrt(0.5));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	elseif (t_m <= 2.95e-5)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(Float64(t_m * cbrt(sin(k))) * cbrt(tan(k))) ^ 3.0)) * Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))) * Float64(l * sqrt(0.5)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.95e-5], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*l* 49.3%

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

      6. associate-*l/ 49.3%

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

      7. associate-*l* 49.3%

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

   6. Applied egg-rr 49.3%

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

   7. Taylor expanded in t around 0 73.4%

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

      1. associate-/l* 73.8%

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

      2. associate-/l* 75.5%

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

   9. Simplified 75.5%

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

   if 4.2e-113 < t < 2.9499999999999999e-5

   1. Initial program 79.6%

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

   3. Simplified 79.5%

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

      1. associate-*r* 83.3%

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

      2. add-sqr-sqrt 83.2%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.6%

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

      1. associate-/l* 91.7%

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

      2. *-commutative 91.7%

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

   8. Simplified 91.7%

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

   9. Taylor expanded in k around inf 63.3%

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

   10. Taylor expanded in k around inf 92.0%

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

   if 2.9499999999999999e-5 < t < 5.60000000000000037e102

   1. Initial program 73.0%

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

   3. Simplified 76.5%

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

      1. associate-*r* 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.3%

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

      4. associate-*r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. *-commutative 87.9%

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

      3. *-commutative 87.9%

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

   8. Simplified 87.9%

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

      1. associate-*r/ 90.2%

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

      2. associate-*l* 90.1%

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

   10. Applied egg-rr 90.1%

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

       1. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

   if 5.60000000000000037e102 < t

   1. Initial program 60.8%

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

   3. Simplified 49.0%

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

      1. associate-*r* 54.4%

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

      2. add-sqr-sqrt 54.4%

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

      3. times-frac 54.4%

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

   6. Applied egg-rr 66.3%

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

      1. associate-/l* 66.3%

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

      2. *-commutative 66.3%

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

   8. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. add-cube-cbrt 66.3%

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

      3. pow3 66.3%

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

      4. cbrt-prod 66.3%

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

      5. unpow3 66.3%

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

      6. add-cbrt-cube 75.2%

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

   10. Applied egg-rr 75.2%

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

       1. add-cube-cbrt 75.1%

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

       2. pow3 75.1%

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

       3. cbrt-prod 75.1%

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

       4. unpow3 75.1%

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

       5. add-cbrt-cube 79.5%

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

   12. Applied egg-rr 79.5%

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

   13. Taylor expanded in k around 0 79.5%

       \[\leadsto \left(\frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \color{blue}{\left(\ell \cdot \sqrt{0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \left(\ell \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{{\left(\left(t\_m \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot t\_2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 4.2e-73)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (*
       t_2
       (* (/ 2.0 (pow (* (* t_m (cbrt (sin k))) (cbrt (tan k))) 3.0)) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 4.2e-73) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else {
		tmp = t_2 * ((2.0 / pow(((t_m * cbrt(sin(k))) * cbrt(tan(k))), 3.0)) * t_2);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 4.2e-73) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else {
		tmp = t_2 * ((2.0 / Math.pow(((t_m * Math.cbrt(Math.sin(k))) * Math.cbrt(Math.tan(k))), 3.0)) * t_2);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 4.2e-73)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	else
		tmp = Float64(t_2 * Float64(Float64(2.0 / (Float64(Float64(t_m * cbrt(sin(k))) * cbrt(tan(k))) ^ 3.0)) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-73], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.1999999999999997e-73

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. associate-*l* 47.2%

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

      2. associate-/r* 51.1%

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

      3. associate-+r+ 51.1%

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

      4. metadata-eval 51.1%

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

      5. associate-*l* 51.1%

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

      6. associate-*l/ 51.6%

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

      7. associate-*l* 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in t around 0 74.1%

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

      1. associate-/l* 74.5%

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

      2. associate-/l* 76.1%

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

   9. Simplified 76.1%

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

   if 4.1999999999999997e-73 < t

   1. Initial program 68.4%

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

   3. Simplified 63.4%

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

      1. associate-*r* 67.3%

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

      2. add-sqr-sqrt 67.2%

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

      3. times-frac 68.3%

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

   6. Applied egg-rr 76.7%

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

      1. associate-/l* 77.9%

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

      2. *-commutative 77.9%

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

   8. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. add-cube-cbrt 77.8%

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

      3. pow3 77.8%

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

      4. cbrt-prod 77.7%

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

      5. unpow3 77.7%

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

      6. add-cbrt-cube 82.4%

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

   10. Applied egg-rr 82.4%

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

       1. add-cube-cbrt 82.3%

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

       2. pow3 82.3%

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

       3. cbrt-prod 82.1%

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

       4. unpow3 82.1%

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

       5. add-cbrt-cube 84.4%

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

   12. Applied egg-rr 84.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t_3 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \frac{\ell}{2 + t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_3\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)) (t_3 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (if (<= t_m 2.2e-6)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (if (<= t_m 5.6e+102)
          (*
           (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0)))
           (/ l (+ 2.0 t_3)))
          (/
           2.0
           (*
            (pow (* (cbrt (sin k)) (* t_m (cbrt (pow l -2.0)))) 3.0)
            (* (tan k) (+ 1.0 (+ 1.0 t_3)))))))))))

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 = (t_m * l) / k;
	double t_3 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else if (t_m <= 2.2e-6) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l / (2.0 + t_3));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m * cbrt(pow(l, -2.0)))), 3.0) * (tan(k) * (1.0 + (1.0 + t_3))));
	}
	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 = (t_m * l) / k;
	double t_3 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else if (t_m <= 2.2e-6) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l / (2.0 + t_3));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m * Math.cbrt(Math.pow(l, -2.0)))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + t_3))));
	}
	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(Float64(t_m * l) / k)
	t_3 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	elseif (t_m <= 2.2e-6)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l / Float64(2.0 + t_3)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m * cbrt((l ^ -2.0)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_3)))));
	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[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-6], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*l* 49.3%

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

      6. associate-*l/ 49.3%

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

      7. associate-*l* 49.3%

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

   6. Applied egg-rr 49.3%

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

   7. Taylor expanded in t around 0 73.4%

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

      1. associate-/l* 73.8%

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

      2. associate-/l* 75.5%

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

   9. Simplified 75.5%

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

   if 4.2e-113 < t < 2.2000000000000001e-6

   1. Initial program 79.6%

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

   3. Simplified 79.5%

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

      1. associate-*r* 83.3%

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

      2. add-sqr-sqrt 83.2%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.6%

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

      1. associate-/l* 91.7%

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

      2. *-commutative 91.7%

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

   8. Simplified 91.7%

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

   9. Taylor expanded in k around inf 63.3%

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

   10. Taylor expanded in k around inf 92.0%

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

   if 2.2000000000000001e-6 < t < 5.60000000000000037e102

   1. Initial program 73.0%

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

   3. Simplified 76.5%

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

      1. associate-*r* 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.3%

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

      4. associate-*r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. *-commutative 87.9%

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

      3. *-commutative 87.9%

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

   8. Simplified 87.9%

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

      1. associate-*r/ 90.2%

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

      2. associate-*l* 90.1%

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

   10. Applied egg-rr 90.1%

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

       1. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

   if 5.60000000000000037e102 < t

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

      1. add-cube-cbrt 60.8%

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

      2. pow3 60.8%

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

      3. associate-/r* 66.3%

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

      4. *-commutative 66.3%

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

      5. cbrt-prod 66.3%

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

      6. associate-/r* 60.8%

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

      7. div-inv 60.8%

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

      8. cbrt-prod 60.8%

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

      9. rem-cbrt-cube 72.7%

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

      10. pow2 72.7%

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

      11. pow-flip 73.6%

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

      12. metadata-eval 73.6%

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

   6. Applied egg-rr 73.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (if (<= t_m 2.95e-5)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (*
         (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
         (* l (/ 2.0 (* (tan k) (pow (* 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 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else if (t_m <= 2.95e-5) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * pow((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 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else if (t_m <= 2.95e-5) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * Math.pow((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(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	elseif (t_m <= 2.95e-5)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(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[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.95e-5], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*l* 49.3%

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

      6. associate-*l/ 49.3%

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

      7. associate-*l* 49.3%

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

   6. Applied egg-rr 49.3%

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

   7. Taylor expanded in t around 0 73.4%

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

      1. associate-/l* 73.8%

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

      2. associate-/l* 75.5%

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

   9. Simplified 75.5%

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

   if 4.2e-113 < t < 2.9499999999999999e-5

   1. Initial program 79.6%

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

   3. Simplified 79.5%

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

      1. associate-*r* 83.3%

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

      2. add-sqr-sqrt 83.2%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.6%

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

      1. associate-/l* 91.7%

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

      2. *-commutative 91.7%

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

   8. Simplified 91.7%

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

   9. Taylor expanded in k around inf 63.3%

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

   10. Taylor expanded in k around inf 92.0%

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

   if 2.9499999999999999e-5 < t

   1. Initial program 65.3%

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

   3. Simplified 59.1%

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

      1. associate-*r* 62.6%

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

      2. *-un-lft-identity 62.6%

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

      3. times-frac 63.9%

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

      4. associate-*r* 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. /-rgt-identity 74.2%

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

      2. *-commutative 74.2%

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

      3. *-commutative 74.2%

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

   8. Simplified 74.2%

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

      1. *-commutative 74.2%

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

      2. add-cube-cbrt 74.2%

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

      3. pow3 74.2%

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

      4. cbrt-prod 74.1%

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

      5. unpow3 74.1%

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

      6. add-cbrt-cube 79.8%

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

   10. Applied egg-rr 79.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (if (<= t_m 8.8e-7)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (if (<= t_m 5.6e+102)
          (*
           (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0)))
           (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
          (*
           (* l (/ 2.0 (* (tan k) (pow (* t_m (cbrt (sin k))) 3.0))))
           (* l 0.5))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else if (t_m <= 8.8e-7) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = (l * (2.0 / (tan(k) * pow((t_m * cbrt(sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else if (t_m <= 8.8e-7) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	elseif (t_m <= 8.8e-7)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (Float64(t_m * cbrt(sin(k))) ^ 3.0)))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.8e-7], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*l* 49.3%

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

      6. associate-*l/ 49.3%

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

      7. associate-*l* 49.3%

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

   6. Applied egg-rr 49.3%

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

   7. Taylor expanded in t around 0 73.4%

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

      1. associate-/l* 73.8%

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

      2. associate-/l* 75.5%

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

   9. Simplified 75.5%

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

   if 4.2e-113 < t < 8.8000000000000004e-7

   1. Initial program 79.6%

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

   3. Simplified 79.5%

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

      1. associate-*r* 83.3%

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

      2. add-sqr-sqrt 83.2%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.6%

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

      1. associate-/l* 91.7%

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

      2. *-commutative 91.7%

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

   8. Simplified 91.7%

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

   9. Taylor expanded in k around inf 63.3%

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

   10. Taylor expanded in k around inf 92.0%

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

   if 8.8000000000000004e-7 < t < 5.60000000000000037e102

   1. Initial program 73.0%

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

   3. Simplified 76.5%

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

      1. associate-*r* 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.3%

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

      4. associate-*r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. *-commutative 87.9%

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

      3. *-commutative 87.9%

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

   8. Simplified 87.9%

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

      1. associate-*r/ 90.2%

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

      2. associate-*l* 90.1%

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

   10. Applied egg-rr 90.1%

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

       1. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

   if 5.60000000000000037e102 < t

   1. Initial program 60.8%

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

   3. Simplified 49.0%

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

      1. associate-*r* 54.4%

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

      2. *-un-lft-identity 54.4%

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

      3. times-frac 54.4%

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

      4. associate-*r* 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. /-rgt-identity 66.3%

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

      2. *-commutative 66.3%

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

      3. *-commutative 66.3%

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

   8. Simplified 66.3%

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

   9. Taylor expanded in k around 0 66.3%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.3%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. *-commutative 66.3%

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

       2. add-cube-cbrt 66.3%

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

       3. pow3 66.3%

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

       4. cbrt-prod 66.3%

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

       5. unpow3 66.3%

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

       6. add-cbrt-cube 75.2%

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

   13. Applied egg-rr 75.1%

       \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}} \cdot \tan k}\right) \cdot \left(\ell \cdot 0.5\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (* l (cos k))))) l))
      (if (<= t_m 3.3e+69)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (*
         (* l (/ 2.0 (* (tan k) (pow (* t_m (cbrt (sin k))) 3.0))))
         (* l 0.5)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / (l * cos(k))))) / l);
	} else if (t_m <= 3.3e+69) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (tan(k) * pow((t_m * cbrt(sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))))) / l);
	} else if (t_m <= 3.3e+69) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))) / l));
	elseif (t_m <= 3.3e+69)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (Float64(t_m * cbrt(sin(k))) ^ 3.0)))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+69], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*l* 49.3%

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

      6. associate-*l/ 49.3%

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

      7. associate-*l* 49.3%

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

   6. Applied egg-rr 49.3%

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

   7. Taylor expanded in t around 0 73.4%

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

      1. associate-/l* 73.8%

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

      2. associate-/l* 75.5%

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

   9. Simplified 75.5%

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

   if 4.2e-113 < t < 3.2999999999999999e69

   1. Initial program 83.7%

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

   3. Simplified 83.6%

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

      1. associate-*r* 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 88.1%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/l* 90.6%

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

      2. *-commutative 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in k around inf 62.6%

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

   10. Taylor expanded in k around inf 86.3%

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

   if 3.2999999999999999e69 < t

   1. Initial program 56.7%

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

   3. Simplified 48.3%

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

      1. associate-*r* 53.0%

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

      2. *-un-lft-identity 53.0%

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

      3. times-frac 54.8%

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

      4. associate-*r* 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. /-rgt-identity 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 67.0%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. *-commutative 68.9%

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

       2. add-cube-cbrt 68.9%

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

       3. pow3 68.9%

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

       4. cbrt-prod 68.8%

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

       5. unpow3 68.8%

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

       6. add-cbrt-cube 76.4%

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

   13. Applied egg-rr 74.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (/ (* (* k k) (* t_m (pow (sin k) 2.0))) (* (cos k) (pow l 2.0))))
      (if (<= t_m 1.35e+69)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (*
         (* l (/ 2.0 (* (tan k) (pow (* t_m (cbrt (sin k))) 3.0))))
         (* l 0.5)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (((k * k) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)));
	} else if (t_m <= 1.35e+69) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (tan(k) * pow((t_m * cbrt(sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (((k * k) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 2.0)));
	} else if (t_m <= 1.35e+69) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 2.0))));
	elseif (t_m <= 1.35e+69)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (Float64(t_m * cbrt(sin(k))) ^ 3.0)))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+69], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in t around 0 67.9%

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

      1. unpow2 67.9%

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

   7. Applied egg-rr 67.9%

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

   if 4.2e-113 < t < 1.3499999999999999e69

   1. Initial program 83.7%

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

   3. Simplified 83.6%

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

      1. associate-*r* 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 88.1%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/l* 90.6%

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

      2. *-commutative 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in k around inf 62.6%

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

   10. Taylor expanded in k around inf 86.3%

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

   if 1.3499999999999999e69 < t

   1. Initial program 56.7%

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

   3. Simplified 48.3%

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

      1. associate-*r* 53.0%

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

      2. *-un-lft-identity 53.0%

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

      3. times-frac 54.8%

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

      4. associate-*r* 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. /-rgt-identity 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 67.0%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. *-commutative 68.9%

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

       2. add-cube-cbrt 68.9%

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

       3. pow3 68.9%

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

       4. cbrt-prod 68.8%

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

       5. unpow3 68.8%

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

       6. add-cbrt-cube 76.4%

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

   13. Applied egg-rr 74.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0)))))
      (if (<= t_m 1.4e+69)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (*
         (* l (/ 2.0 (* (tan k) (pow (* t_m (cbrt (sin k))) 3.0))))
         (* l 0.5)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 1.4e+69) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (tan(k) * pow((t_m * cbrt(sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 1.4e+69) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0)))) * (l * 0.5);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 1.4e+69)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (Float64(t_m * cbrt(sin(k))) ^ 3.0)))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+69], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in t around 0 67.9%

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

   6. Taylor expanded in k around 0 61.9%

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

      1. div-inv 61.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 62.4%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 62.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 62.4%

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

   10. Simplified 62.4%

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

   if 4.2e-113 < t < 1.39999999999999991e69

   1. Initial program 83.7%

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

   3. Simplified 83.6%

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

      1. associate-*r* 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 88.1%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/l* 90.6%

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

      2. *-commutative 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in k around inf 62.6%

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

   10. Taylor expanded in k around inf 86.3%

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

   if 1.39999999999999991e69 < t

   1. Initial program 56.7%

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

   3. Simplified 48.3%

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

      1. associate-*r* 53.0%

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

      2. *-un-lft-identity 53.0%

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

      3. times-frac 54.8%

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

      4. associate-*r* 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. /-rgt-identity 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 67.0%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. *-commutative 68.9%

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

       2. add-cube-cbrt 68.9%

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

       3. pow3 68.9%

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

       4. cbrt-prod 68.8%

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

       5. unpow3 68.8%

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

       6. add-cbrt-cube 76.4%

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

   13. Applied egg-rr 74.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right) \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m l) k)))
   (*
    t_s
    (if (<= t_m 4.2e-113)
      (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0)))))
      (if (<= t_m 2.8e+34)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (* (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0))) (* l 0.5)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 2.8e+34) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l * 0.5);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m * l) / k
    if (t_m <= 4.2d-113) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    else if (t_m <= 2.8d+34) then
        tmp = t_2 * ((2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0)))) * t_2)
    else
        tmp = (((2.0d0 * l) / sin(k)) / (tan(k) * (t_m ** 3.0d0))) * (l * 0.5d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * l) / k;
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 2.8e+34) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l * 0.5);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m * l) / k
	tmp = 0
	if t_m <= 4.2e-113:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	elif t_m <= 2.8e+34:
		tmp = t_2 * ((2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0)))) * t_2)
	else:
		tmp = (((2.0 * l) / math.sin(k)) / (math.tan(k) * math.pow(t_m, 3.0))) * (l * 0.5)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * l) / k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 2.8e+34)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m * l) / k;
	tmp = 0.0;
	if (t_m <= 4.2e-113)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	elseif (t_m <= 2.8e+34)
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0)))) * t_2);
	else
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * (t_m ^ 3.0))) * (l * 0.5);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e+34], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in t around 0 67.9%

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

   6. Taylor expanded in k around 0 61.9%

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

      1. div-inv 61.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 62.4%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 62.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 62.4%

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

   10. Simplified 62.4%

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

   if 4.2e-113 < t < 2.80000000000000008e34

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

      1. associate-*r* 87.8%

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

      2. add-sqr-sqrt 87.6%

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

      3. times-frac 90.7%

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

   6. Applied egg-rr 90.9%

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

      1. associate-/l* 93.9%

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

      2. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in k around inf 61.4%

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

   10. Taylor expanded in k around inf 88.4%

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

   if 2.80000000000000008e34 < t

   1. Initial program 60.0%

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

   3. Simplified 52.9%

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

      1. associate-*r* 56.9%

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

      2. *-un-lft-identity 56.9%

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

      3. times-frac 58.4%

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

      4. associate-*r* 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. /-rgt-identity 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

   8. Simplified 70.3%

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

   9. Taylor expanded in k around 0 67.2%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.2%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 67.2%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 71.3%

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

       2. associate-*l* 71.3%

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

   13. Applied egg-rr 68.1%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\sin k \cdot \left({t}^{3} \cdot \tan k\right)}} \cdot \left(\ell \cdot 0.5\right) \]
   14. Step-by-step derivation

       1. associate-/r* 75.0%

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

   15. Simplified 71.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \ell}{k} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot \frac{t\_m \cdot \ell}{k}\right) \cdot \left(\ell \cdot \frac{t\_m}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-113)
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0)))))
    (if (<= t_m 3.7e+34)
      (*
       (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) (/ (* t_m l) k))
       (* l (/ t_m k)))
      (* (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0))) (* l 0.5))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 3.7e+34) {
		tmp = ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * ((t_m * l) / k)) * (l * (t_m / k));
	} else {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l * 0.5);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-113) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    else if (t_m <= 3.7d+34) then
        tmp = ((2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0)))) * ((t_m * l) / k)) * (l * (t_m / k))
    else
        tmp = (((2.0d0 * l) / sin(k)) / (tan(k) * (t_m ** 3.0d0))) * (l * 0.5d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-113) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 3.7e+34) {
		tmp = ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * ((t_m * l) / k)) * (l * (t_m / k));
	} else {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l * 0.5);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-113:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	elif t_m <= 3.7e+34:
		tmp = ((2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0)))) * ((t_m * l) / k)) * (l * (t_m / k))
	else:
		tmp = (((2.0 * l) / math.sin(k)) / (math.tan(k) * math.pow(t_m, 3.0))) * (l * 0.5)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-113)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 3.7e+34)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * Float64(Float64(t_m * l) / k)) * Float64(l * Float64(t_m / k)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l * 0.5));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-113)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	elseif (t_m <= 3.7e+34)
		tmp = ((2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0)))) * ((t_m * l) / k)) * (l * (t_m / k));
	else
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * (t_m ^ 3.0))) * (l * 0.5);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-113], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.7e+34], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-113

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in t around 0 67.9%

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

   6. Taylor expanded in k around 0 61.9%

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

      1. div-inv 61.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 62.4%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 62.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 62.4%

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

   10. Simplified 62.4%

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

   if 4.2e-113 < t < 3.70000000000000009e34

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

      1. associate-*r* 87.8%

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

      2. add-sqr-sqrt 87.6%

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

      3. times-frac 90.7%

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

   6. Applied egg-rr 90.9%

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

      1. associate-/l* 93.9%

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

      2. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in k around inf 61.4%

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

   10. Taylor expanded in k around inf 88.4%

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

       1. associate-/l* 85.5%

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

   12. Simplified 85.5%

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

   if 3.70000000000000009e34 < t

   1. Initial program 60.0%

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

   3. Simplified 52.9%

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

      1. associate-*r* 56.9%

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

      2. *-un-lft-identity 56.9%

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

      3. times-frac 58.4%

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

      4. associate-*r* 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. /-rgt-identity 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

   8. Simplified 70.3%

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

   9. Taylor expanded in k around 0 67.2%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.2%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 67.2%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 71.3%

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

       2. associate-*l* 71.3%

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

   13. Applied egg-rr 68.1%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\sin k \cdot \left({t}^{3} \cdot \tan k\right)}} \cdot \left(\ell \cdot 0.5\right) \]
   14. Step-by-step derivation

       1. associate-/r* 75.0%

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

   15. Simplified 71.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{t \cdot \ell}{k}\right) \cdot \left(\ell \cdot \frac{t}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e+54)
    (* (/ (/ (* 2.0 l) (sin k)) (* (tan k) (pow t_m 3.0))) (* l 0.5))
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e+54) {
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * pow(t_m, 3.0))) * (l * 0.5);
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.35d+54) then
        tmp = (((2.0d0 * l) / sin(k)) / (tan(k) * (t_m ** 3.0d0))) * (l * 0.5d0)
    else
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e+54) {
		tmp = (((2.0 * l) / Math.sin(k)) / (Math.tan(k) * Math.pow(t_m, 3.0))) * (l * 0.5);
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.35e+54:
		tmp = (((2.0 * l) / math.sin(k)) / (math.tan(k) * math.pow(t_m, 3.0))) * (l * 0.5)
	else:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e+54)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / Float64(tan(k) * (t_m ^ 3.0))) * Float64(l * 0.5));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.35e+54)
		tmp = (((2.0 * l) / sin(k)) / (tan(k) * (t_m ^ 3.0))) * (l * 0.5);
	else
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.35e+54], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000005e54

   1. Initial program 58.6%

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

   3. Simplified 53.9%

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

      1. associate-*r* 58.3%

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

      2. *-un-lft-identity 58.3%

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

      3. times-frac 57.1%

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

      4. associate-*r* 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. /-rgt-identity 62.6%

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

      2. *-commutative 62.6%

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

      3. *-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in k around 0 65.4%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 65.4%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 62.9%

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

       2. associate-*l* 62.9%

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

   13. Applied egg-rr 65.7%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\sin k \cdot \left({t}^{3} \cdot \tan k\right)}} \cdot \left(\ell \cdot 0.5\right) \]
   14. Step-by-step derivation

       1. associate-/r* 64.4%

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

   15. Simplified 67.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot 2}{\sin k}}{{t}^{3} \cdot \tan k}} \cdot \left(\ell \cdot 0.5\right) \]

   if 1.35000000000000005e54 < k

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 72.3%

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

   6. Taylor expanded in k around 0 64.8%

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

      1. div-inv 64.8%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 66.3%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 66.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{\tan k \cdot {t}^{3}} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+53}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.55e+53)
    (* (* l 0.5) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.55e+53) {
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.55d+53) then
        tmp = (l * 0.5d0) * (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0)))))
    else
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.55e+53) {
		tmp = (l * 0.5) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.55e+53:
		tmp = (l * 0.5) * (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0)))))
	else:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.55e+53)
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.55e+53)
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0)))));
	else
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.55e+53], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.54999999999999987e53

   1. Initial program 58.6%

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

   3. Simplified 53.9%

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

      1. associate-*r* 58.3%

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

      2. *-un-lft-identity 58.3%

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

      3. times-frac 57.1%

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

      4. associate-*r* 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. /-rgt-identity 62.6%

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

      2. *-commutative 62.6%

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

      3. *-commutative 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in k around 0 65.4%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 65.4%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 3.54999999999999987e53 < k

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 72.3%

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

   6. Taylor expanded in k around 0 64.8%

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

      1. div-inv 64.8%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 66.3%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 66.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+53}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{4}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-34)
    (* (* l 0.5) (* l (/ 2.0 (* (tan k) (* k (pow t_m 3.0))))))
    (/ 2.0 (/ (* t_m (pow k 4.0)) (* (cos k) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / ((t_m * pow(k, 4.0)) / (cos(k) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-34) then
        tmp = (l * 0.5d0) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    else
        tmp = 2.0d0 / ((t_m * (k ** 4.0d0)) / (cos(k) * (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = (l * 0.5) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k, 4.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.25e-34:
		tmp = (l * 0.5) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 3.0)))))
	else:
		tmp = 2.0 / ((t_m * math.pow(k, 4.0)) / (math.cos(k) * math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.25e-34)
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k ^ 4.0)) / Float64(cos(k) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.25e-34)
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	else
		tmp = 2.0 / ((t_m * (k ^ 4.0)) / (cos(k) * (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.25e-34], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2500000000000001e-34

   1. Initial program 60.8%

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

   3. Simplified 55.5%

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

      1. associate-*r* 59.3%

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

      2. *-un-lft-identity 59.3%

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

      3. times-frac 57.9%

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

      4. associate-*r* 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. /-rgt-identity 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in k around 0 66.7%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.7%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   12. Taylor expanded in k around 0 65.4%

       \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \tan k}\right) \cdot \left(\ell \cdot 0.5\right) \]

   if 1.2500000000000001e-34 < k

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 72.1%

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

   6. Taylor expanded in k around 0 63.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.15e-34)
    (* (* l 0.5) (* l (/ 2.0 (* (tan k) (* k (pow t_m 3.0))))))
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-34) {
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d-34) then
        tmp = (l * 0.5d0) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    else
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-34) {
		tmp = (l * 0.5) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.15e-34:
		tmp = (l * 0.5) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 3.0)))))
	else:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.15e-34)
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.15e-34)
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	else
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.15e-34], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.15000000000000006e-34

   1. Initial program 60.8%

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

   3. Simplified 55.5%

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

      1. associate-*r* 59.3%

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

      2. *-un-lft-identity 59.3%

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

      3. times-frac 57.9%

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

      4. associate-*r* 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. /-rgt-identity 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in k around 0 66.7%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.7%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   12. Taylor expanded in k around 0 65.4%

       \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \tan k}\right) \cdot \left(\ell \cdot 0.5\right) \]

   if 1.15000000000000006e-34 < k

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 72.1%

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

   6. Taylor expanded in k around 0 63.4%

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

      1. div-inv 63.4%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* 64.5%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

   8. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.5%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot \frac{t}{{\ell}^{2} \cdot \cos k}}} \]

      2. metadata-eval 64.5%

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

   10. Simplified 64.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.95 \cdot 10^{+40}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.95e+40)
    (* (* l 0.5) (* l (/ 2.0 (* (tan k) (* k (pow t_m 3.0))))))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.95e+40) {
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.95d+40) then
        tmp = (l * 0.5d0) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.95e+40) {
		tmp = (l * 0.5) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.95e+40:
		tmp = (l * 0.5) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 3.0)))))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.95e+40)
		tmp = Float64(Float64(l * 0.5) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.95e+40)
		tmp = (l * 0.5) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.95e+40], N[(N[(l * 0.5), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.9499999999999999e40

   1. Initial program 58.7%

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

   3. Simplified 54.0%

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

      1. associate-*r* 58.4%

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

      2. *-un-lft-identity 58.4%

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

      3. times-frac 57.1%

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

      4. associate-*r* 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. /-rgt-identity 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

   8. Simplified 62.7%

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

   9. Taylor expanded in k around 0 65.6%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.6%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 65.6%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   12. Taylor expanded in k around 0 64.0%

       \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \tan k}\right) \cdot \left(\ell \cdot 0.5\right) \]

   if 3.9499999999999999e40 < k

   1. Initial program 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around 0 71.6%

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

   6. Taylor expanded in k around 0 62.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.95 \cdot 10^{+40}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.5e+81)
    (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k k))))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e+81) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d+81) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k * k)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e+81) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.5e+81:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.5e+81)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.5e+81)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k * k)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.5e+81], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5e81

   1. Initial program 57.5%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in k around 0 58.7%

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

      1. unpow2 61.9%

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

   7. Applied egg-rr 58.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. associate-/r* 29.8%

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

      4. sqrt-div 29.8%

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

      5. sqrt-pow1 29.9%

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

      6. metadata-eval 29.9%

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

      7. sqrt-prod 15.5%

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

      8. add-sqr-sqrt 31.7%

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

   9. Applied egg-rr 31.7%

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

   if 3.5e81 < k

   1. Initial program 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in t around 0 68.3%

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

   6. Taylor expanded in k around 0 64.1%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \frac{2 \cdot \ell}{{k}^{2} \cdot {t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-34)
    (* (* l 0.5) (/ (* 2.0 l) (* (pow k 2.0) (pow t_m 3.0))))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = (l * 0.5) * ((2.0 * l) / (pow(k, 2.0) * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-34) then
        tmp = (l * 0.5d0) * ((2.0d0 * l) / ((k ** 2.0d0) * (t_m ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = (l * 0.5) * ((2.0 * l) / (Math.pow(k, 2.0) * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.25e-34:
		tmp = (l * 0.5) * ((2.0 * l) / (math.pow(k, 2.0) * math.pow(t_m, 3.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.25e-34)
		tmp = Float64(Float64(l * 0.5) * Float64(Float64(2.0 * l) / Float64((k ^ 2.0) * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.25e-34)
		tmp = (l * 0.5) * ((2.0 * l) / ((k ^ 2.0) * (t_m ^ 3.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.25e-34], N[(N[(l * 0.5), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2500000000000001e-34

   1. Initial program 60.8%

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

   3. Simplified 55.5%

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

      1. associate-*r* 59.3%

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

      2. *-un-lft-identity 59.3%

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

      3. times-frac 57.9%

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

      4. associate-*r* 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. /-rgt-identity 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in k around 0 66.7%

      \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.7%

       \[\leadsto \left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   12. Taylor expanded in k around 0 60.2%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot {t}^{3}}\right)} \cdot \left(\ell \cdot 0.5\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 60.2%

          \[\leadsto \color{blue}{\frac{2 \cdot \ell}{{k}^{2} \cdot {t}^{3}}} \cdot \left(\ell \cdot 0.5\right) \]

       2. *-commutative 60.2%

          \[\leadsto \frac{\color{blue}{\ell \cdot 2}}{{k}^{2} \cdot {t}^{3}} \cdot \left(\ell \cdot 0.5\right) \]

   14. Simplified 60.2%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{{k}^{2} \cdot {t}^{3}}} \cdot \left(\ell \cdot 0.5\right) \]

   if 1.2500000000000001e-34 < k

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 72.1%

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

   6. Taylor expanded in k around 0 59.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \frac{2 \cdot \ell}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-34)
    (/ 2.0 (* (* 2.0 (* k k)) (/ (* t_m (/ (pow t_m 2.0) l)) l)))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (pow(t_m, 2.0) / l)) / l));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-34) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.25e-34:
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.25e-34)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.25e-34)
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * ((t_m ^ 2.0) / l)) / l));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.25e-34], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2500000000000001e-34

   1. Initial program 60.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in k around 0 59.2%

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

      1. unpow2 59.1%

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

   7. Applied egg-rr 59.2%

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

      1. associate-/r* 56.2%

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

      2. unpow3 56.2%

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

      3. times-frac 60.4%

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

      4. pow2 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate-*r/ 60.4%

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

   11. Applied egg-rr 60.4%

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

   if 1.2500000000000001e-34 < k

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 72.1%

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

   6. Taylor expanded in k around 0 59.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.1% accurate, 3.6× speedup

Math

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

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 (* (* 2.0 (* k k)) (/ (* t_m (/ (pow t_m 2.0) 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m * (pow(t_m, 2.0) / l)) / 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 / ((2.0d0 * (k * k)) * ((t_m * ((t_m ** 2.0d0) / l)) / 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 / ((2.0 * (k * k)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / 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 / ((2.0 * (k * k)) * ((t_m * (math.pow(t_m, 2.0) / l)) / 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(2.0 * Float64(k * k)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / 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 / ((2.0 * (k * k)) * ((t_m * ((t_m ^ 2.0) / l)) / 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[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Simplified 55.6%

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

5. Taylor expanded in k around 0 56.9%

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

   1. unpow2 63.1%

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

7. Applied egg-rr 56.9%

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

   1. associate-/r* 53.9%

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

   2. unpow3 53.9%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

    1. associate-*r/ 58.1%

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

11. Applied egg-rr 58.1%

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

12. Final simplification 58.1%

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

Alternative 20: 58.4% accurate, 24.8× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ (* t_m t_m) l) (/ t_m 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 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / 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 / ((2.0d0 * (k * k)) * (((t_m * t_m) / l) * (t_m / 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 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / 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 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / 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(2.0 * Float64(k * k)) * Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / 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 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / 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[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Simplified 55.6%

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

5. Taylor expanded in k around 0 56.9%

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

   1. unpow2 63.1%

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

7. Applied egg-rr 56.9%

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

   1. associate-/r* 53.9%

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

   2. unpow3 53.9%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

    1. unpow2 58.1%

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

11. Applied egg-rr 58.1%

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

12. Final simplification 58.1%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))